I think his parents were probably a Henry and Mary too but don't think Mary's surname was Thick as shown on the various trees. That marriage took place in London and trees show she died St. Pancras in 1840 when the Cope family were fully ensconced in Essex!!
Between 1798 and 1818 a Henry and Mary had children baptised Romford, Dagenham and Barking and in 1841 Henry is in Barking (no wife with him) and living with him are his married daughter Maria (Flack) and her family. Henry Cope bc.1768 buried 21/6/1846 Gt. Ilford. Henry and Mary had a dau. Phoebe Ann bp.1804 Dagenham (not a common name for the time) and I note that William and Sarah named 2 of their daughters Maria and Phoebe.
Annette