In the 1900 census he appears as 49 years old, immigrated in 1886, married in 1877, wife Anna (age 55), children Carl (22), Emma (21), Willie (19), and Fred (16) born in Germany, daughter Ellen (11) born in New York (see ancestry.com behind paywall).
On this basis he can be identified as the Hans Chr. (Christian) Thiede, age 35, who departed on the Rhaetia from Hamburg on 4 March 1886 with wife Anna (age 41) and children Carl (9), Emma (7), Wilhelm (4), and Friedrich (3) (see Hamburg shipping list at ancestry.com). This Hans Christian Thiede gave his place of origin as Appen in Holstein (just outside Pinneberg, about 20 km northwest of Hamburg).
O.k., I am afraid I'm coming up empty. You are not going to find their parents at ancestry.com anyway, as ancestry.com does not have the church records from that area of Holstein.
Appen was in the clerical district ("Kirchspiel") of Rellingen. The church books for Rellingen are indeed available at archion.de (a paysite run by the Protestant churches of Germany). There are no indexes available for baptisms, but I thumbed through the baptismal registers for late 1850/early 1851 and did not find an entry for a Tiede/Thiede there. Perhaps another German researcher who has access to archion.de might want to doublecheck my work here.
It also appears that this area was included in the Danish census of 1855. The volume for Pinneberg district ("Amt Pinneberg"), in which Appen was located, has been put online by familysearch.org (which is a free site, but requires registration) here. Unfortunately, this particular area appears to be one of the few parts of Holstein whose censuses have not yet been indexed by the Danes (for their census indexes, which are also free, see here). And quite frankly I don't feel like thumbing through 600 pages of a census volume to try and find this family - though someone else might want to give it a try.