Sorry Kenneth, but I think you have two different Beardsley families intertwined; one from Keele Staffordshire, and the other from St Albans, Hertforshire, and then if Richard is from Essex, then this might represent a third family? Unless they are from the gentry where such moving around might be understandable, people didn't move around that much, and if they were gentry you might expect to find more on the family?
Aside from that, Thomas Beardsley is really at the uppermost limit of where you are going to find records easily. Before that it would only be if there were some sort of legal record that names his parents.
However Findmypast does seem to have parish records from St Albans and it might be worth asking if someone has access can do some lookups for you, though looking at the Index, nothing positive about his parents was evident.
He may have married twice (not sure if this link will work) but it came from Hertfordshire Names Online. Joane Upton and Joane Meret
He does have a non-cupative will and inventory, but it looks like you might have to apply to get it and not sure how much that will cost. This might tell you something?