Where was Francis born? The link on the profile only goes to Ancestry CA trees. I couldn't access them but searched from them on Ancestry UK. As far as I could see they have a dob but neither a place nor a source for the information. Ancestry actually has very few records for Wiltshire parishes other than indexed marriages.
As I mentioned before, there also aren't registers this early for quite a lot of the parishes (they may have got lost or damaged by damp and vermin).
These are the parish registers available for Wiltshire http://www.wshc.eu/parish-register-list.html (notice that Brinkworth where John and Francis lived has registers going back to only 1653 so unless he was baptised elsewhere, the information on his baptism may not be in any existing record)
Some, registers, but by no means all are transcribed and are freely available on the Wiltshire OPC website. http://www.wiltshire-opc.org.uk/genealogy/index.php/parish-directory
Some of the parishes on the OPC, including Brinkworth have transcriptions of other early records which can sometimes help take things further back.
One thing that probably won't help find his parent but may be interesting isthis inventory A list of his possessions taken after his death and the person who was allowed to deal with his estate
Ref No. P3/S/437 Title Administration bond, inventory Date1667
Person Spencer, Francis Occupation Yeoman
Place Key /Grittenham/Brinkworth/Wiltshire Community Brinkworth
(you would have to get it copied and sent to you from the archives. Unfortunately this one is not online) http://history.wiltshire.gov.uk/heritage/getwill.php?id=72857
The burial of Francis Spencer burial took place on 8th August, 1667
Brinkworth Burial Register 1663-1699, transcription http://www.wiltshire-opc.org.uk/genealogy/index.php/parish-directory/item/324-brinkworth