How correct is this sentence: "His parents did not approve of our dating because of our age difference, but we did anyway."
The sentences are runny and confusing so I would reword as this:
His parents didn't approve of the two of us dating because of the difference in our age. We are dating regardless.
The second sentence is also ambiguous so I wouldn't use it in that structure.
My sister hates swimming but I do not. I swim everyday.