Is it permissible for my father to miss Friday prayer due to bad breath and its foul odor that might harm other worshippers?

Yes, a very strong foul odor from bad breath or body odor, if the person cannot remove it, is considered a legitimate excuse that prevents one from attending congregational prayers (jama'ah) and Friday prayers (Jumu'ah). This is a stronger excuse than that of someone who has eaten onions, because the latter can refrain from eating onions. However, if these odors can be removed through hygiene or medication, then it is obligatory to remove them in order to attend prayer. This also applies to the smell of smoke; a smoker must remove the trace of smoke from their mouth before going to the mosque, and they should strive to quit smoking because it is an abomination.