It is uncertain at this time as to the identity of Henry's parents. However, Y DNA does indicate an NPE in this line as male descendants of Henry Franklin Sr. who have taken a Y-DNA test match various members of the Long family of Virginia, including a Broomfield Long and his possible grandson, John Long.
There is also confusion over Henry's maternal line. Miss Mathis, possible daughter of Benjamin Mathis, vies with Mary Payne, and it may be that Lawrence Franklin had two wives, a Mathis and a Payne. This question needs commentary by someone who has researched this, but as Lawrence and Edward resided in VA well before 1700, it may be that sufficient records establishing familial connections simply don't exist.
It is likely that Henry Franklin and his brother Reuben Franklin may have been adopted by Lawrence Franklin after the death of their parents Henry Long Jr. and Ann LNU of Essex/Spotsylvania County. Henry Long Jr. was born about 1685-90, and was the son of Henry Long Sr. and FNU Churchill. Broomfield Long was the brother of Henry Long Jr. Henry Long Jr. and his wife died probably died in the 1720s or early 1730s, and their children were probably adopted out. Lawrence Franklin may have adopted Henry Long and Reuben Long, and they eventually became known as Henry Franklin and Reuben Franklin.
 The key reference is on page 19 where the author states: “Henry's wife’s name, Ann____, turns up in a land transaction in 1738, when the land they had been living on was sold [as well a piece of land given Henry by his father in 1716]. There are no further records pertaining to Henry and Ann. It has been found that a “Franklyn” descendant’s DNA matches the Henry Long line. Court records [Spotsylvania 1737/38] have turned up a case with Henry M. Franklin b. 1715 in Essex county cited as aka “Harry Long”, who would have been 21 at the time. Speculating this to have been Henry’s son, it suggests “Harry’s” parents had died and Harry was adopted by the Franklyns. The land sold in 1738 was most likely his inheritance from his father, Henry, which he sold when he reached majority."
In about 1752 in Orange county, Virginia, Henry married Margaret Hensley.  They had the following children: 
1735 Henry's adopted parents Lawrence Franklin and Mary Payne move to Orange county. They travel back and forth between Orange and Sppotsylvania counties over the next two to years handling business matters (records appear for them in the courts of both counties).
25 July 1737 First court record for young Henry. He had probably just turned 21. Henry Franklin alias Henry Long is mentioned as a juror or witness in a case between John King and Larkin Chew in Spotsylvania County, Virginia.
7 February 1738 Defendant in Trepass Case in Spotsylvania County. Multiple court records/hearings in 1738 in the Trespass action between MARY CURTIS an Infant by RICE CURTIS JUNR. Gent.. her next friend, against HENRY FRANKLYN alias HARRY LONG Deft. for Ten pounds Sterling damage.
About 1740 Henry moves to Orange county to live near his adopted parents Lawrence Franklin and Mary Payne. He begins to appear on Road Repair Orders in Orange County with his adopted father Lawrence.
29 June 1750 Henry selected as a juror in a trial in Orange County.
28 March 1751 Henry is appointed overseer in a Road Order in Orange County.
21 August 1751 Henry is asked to determine the value of the estate of John McKinley, deceased, in Orange County.
28 February 1752 Henry selected as a juror in a trial in Orange County.
25 November 1752 Henry is a witness in a lawsuit between Thomas Walker, Gentleman, and William Watkins in Orange County.
26 April 1753 Henry is the defendant in an assault and battery suit in Orange County.
6 April 1753 - Henry is the plaintiff in a lawsuit for trespass assault against Henry Cave in Orange County.
25 May 1753 Henry selected as a juror in a trial in Orange County. In the same court session, Henry is the defendant in a lawsuit for trespass in Orange County.
Sometime between 1753-1765, Henry moved from Orange County to neighboring Albemarle County.
10 Apr 1765: Henry purchased 156 acres on a branch of Beaver Creek in Albemarle County for £42 from [son-in-law] Joseph and [Joseph's brother] Richard Ballenger. Witnesses included [sons] James and Henry Franklin Junior. On that same day, along with Drury Tucker, Henry witnessed a deed from Joseph Ballenger to Richard Ballenger in Amherst County, VA, also on a branch of Beaver Creek.
4 Mar 1771: Henry Franklin Senior witnessed a deed of Joseph Ballenger, along with [son] John Franklin and John Smith in Albemarle County.
About this time, Henry may have moved to nearby Amherst County.
25 Mar 1771: Henry Franklin Senior witnessed a deed of Joseph Ballenger, and was listed in that deed as the owner of adjoining land on Smith's Mountain in Amherst County.
1783 Tax List, Amherst County, VA: 6 whites; 2 blacks. No Township Listed. 
17 Nov 1784: He gave consent for the marriage of [daughter] Philadelphia Franklin to William Smith in Amherst County, with [son] Samuel Franklin as surety.
10 Apr 1792: Henry signed his will . . .
Will of Henry Franklin
I, Henry Franklin of Amherst County, being weak in body [but] of sound mind and memory, do make and ordain this my last will and testament and first I give my soul to God and my body I recommend to the earth to be buried at the discretion of my Executors hereafter to be named, and as touching my worldly estate I give devise and dispose of the same in the manner and form following. I give to my son-in-law Joseph Ballinger all the commodities I lent him some time ago, to wit, one dun colored mare with her increase, some cows, feather bed that he has now in possession. I give to my son John Franklin one dun horse, one cow and calf & her increase with one feather bed, to which things he has now in his possession. I give to my daughter Elizabeth Ballinger one dun horse and saddle and cow and calf with her increase & one feather bed, to which things she has now in possession. I give to my son Henry Franklin one brown coloured horse, one cow and calf & their increase and one feather bed, to which things he has in his possession. I give to my son James Franklin one feather bed and furniture, which things he has not in his possession. I give to my daughter Peggy Harrison one dun coloured mare and saddle and feather bed and one cow and calf and their increase, which things she has in her possession. I give to my daughter Phebe Franklin one feather bed and furniture, one woman's riding saddle & one cow and calf and their increase, which things she has in her possession. I give to my son Joel Franklin one black horse colt, one feather bed and furniture and one cow and calf which he has in possession. I give to my son Samuel Franklin my tract of land whereon I now live to be freely possessed and enjoyed by him at the decease of his mother. I give to my daughter Philadelphia Smith one dun coloured mare colt likewise what other things I have heretofore given her which she has in her possession. All the remainder of my estate after paying my just debts I give to my beloved wife Margaret Franklin during her natural life and at her decease it is my will and desire that the said estate so remaining be equally divided between my son Samuel Franklin and my daughter Judith Franklin. And lastly I constitute and appoint my two sons, viz Joel Franklin and Samuel Franklin, Executors to this my last will and testament, and do hereby disannul every other wills and legacies & Executors, ratifying and confirming this and no other to be my last will and testament. Given under my hand and seal this tenth day of April one thousand seven hundred and ninety-two.
Henry Franklin (Seal)
Signed, sealed and pronounced In the presence of
Andrew (x) Harrison
Wife Margaret was obviously still alive in 1792 when named in the above will, although one anonymous sources claims her death date was 30 Apr 1788.
17 Sep 1792: The will was proved by Philip Smith and Aaron Franklin. Executors Joel and Samuel Franklin posted bond of £500, with William Ware as security.
October 1792: The estate inventory was filed by Samuel Franklin, the appraisers being Richard Oglesby, John Sale, John Brockman, and Leonard Henley. Is "Henley" actually "Hensley"?
Note that child Judith is not mentioned in Henry's will. Is she really one of his children? 
Henry Franklin was in Revolutionary War Virginia, 2nd Regiment, 1777-80 
↑ Article from The Virginia Genealogist, Vol. 38, No. 2 (Apr.-Jun. 1994):115-126, “Henry Long and some descendants of Colonial Virginia" by Paul Buchanan. Also the book, LONG Journey West By JL Brannon.