'why do they exist' is kind of a strange question; why does any synonym exist? If you're asking for a 'difference' between the two expressions, then 父母 is a biological term implicating a biological mother and father, and 両親 is a social term-- an adopted child's parents are 両親, not 父母. Naturally, a non-adopted child's parents are 両親 as well.
As for 母親/父親, these words ring a bit more 'proper' than お父さん/お母さん. In this case, it's a bit more about nuance-- the latter is a bit warmer/intimate than the former. You might directly call your father お父さん, but when talking to your boss about him, it would likely be preferable to call him 父親.