If you Father married bigamously then he never actually married, so no certificate should be issued because it should be marked as such.
There is such a process, at least in theory, but I have numerous certificates from marriages known to be bigamous (and with convictions at court) and I've yet to have one refused, or see a register entry that has been annotated as bigamous, though no doubt they do exist.
I have a copy of a note from the Registrar General in papers relating to a bigamous marriage about whether such an entry should be removed from a register (in series RG48 at TNA) where it is stated that:
A marriage entry is not a guarantee that a valid marriage subsists between the parties specified, it is a record of the fact that a marriage was celebrated between certain persons after fulfilment of all the requirements of the law.Which suggests that the RG was not that concerned about later issues about the legality of a marriage - just recording the event that took place on the day.