nelliedee, I’m assuming that there wasn’t any information given about parents in the Alfreton marriage register in 1811? Did your elderly relative give his reason for thinking that the family came from Ashover, other than the fact that there were a lot of them there? Also, I'm not quite sure why you've looked in the parish registers for Matlock when he lived in Swanwick and got married in Alfreton. Have you checked the Alfreton registers? Sorry for being thick here!
I still think it would be worth your while getting his death certificate…is it the one listed in 1848 in Belper? If you’ve got a better idea of his year of birth, then you can rule out quite a lot of your possibles and narrow it down considerably.
What was Henry’s trade? I recently found out who the father of my 4x great grandfather was from an entry I found on Google Books in the list of the Freemen of Great Yarmouth, which also listed the fathers or apprentice masters. That was an amazing find! If your ancestor was a member of a guild or similar organisation, it might say who his father was.
Otherwise you could try and see if he was named in a will, which is a way I found out about the parents of one of my 5x great grandfathers. Do you know the National Archives website? You could search there, although they only have wills up to 1859 (I think).
http://www.nationalarchives.gov.uk/documentsonline/power-search.asp?searchType=powersearch I’ve just done a quick search for Handbury and a will came up for John Handbury or Hanbury 1650 in Worcester, so not much help. But you could search through the probate indexes on Find My Past if it seems likely that Henry’s father might have left a will. It’s going to be complicated by the fact that you won’t know his first name or when he died, but if you find any possibles it might be worth taking a punt and sending off for the wills.
In case you don’t know how to find the Probate Indexes on FindMyPast, click on this link below:
http://www.findmypast.com/BirthsMarriagesDeaths.jsp and then look down the Deaths and Burials bit until you find ‘Index to death duty registers 1796-1903’. You have to search them year by year, and if you don’t have a subscription (as I don’t) you have to buy credits and it can get a bit pricey. You can go to London and search for wills personally, or you can pay Her Majesty’s Courts Service to look for you. More details here:
http://www.hmcourts-service.gov.uk/cms/1183.htm I’m not a huge authority on wills, though. Maybe someone who knows more about it than I do knows a better way.
Anyway, there are a few suggestions to be going on with. Try to think outside the box, what sort of forms do we fill in today that require our parents’ names? Good luck!