I looked it up in my monolingual dictionary and I got the following:
Which seems pretty straightforward, but then I got this:
As you can see it doesn’t give a real explanation on the difference in usage between the two.
I know the difference between 親 and 両親 however. The first just refers to (a) parent(s), while the second refers to both parents.
Is their a difference in nuance like 二親 putting stress on the fact that you have (to consider) both parents?
Dictionary app I used