Once yuo get back pre1837 you are more reliant on Parish registers.
These are not often on line, though you may find some entries on the IGI (searchable free at familysearch.org, "extracted" entries, although best checked out, are worth looking at, "submitted" entries are sometimes no more than someones imagination!) Different area have better coverage than others on the IGI
Local Record Offices will hold Parish registers, I dont know where you live, but it always fun to go and search through them yourself if yuo can - for christeneings, burilas and marriages.
Otherwise, you may find a local rootschatter who has access to them.
Do you know if Henry and Ann married after 1837? (d-o-b for the children might give an indication)?
If so, a marriage cert will give father's name and occupation and therefore be a first step to going back.
Have you got them on any censsues?