How many pairs of parallel diagonals in an octagon?

This assumes you mean a regular octagon.

There are 4 pairs where the diagonals each skip two vertices.

Then there are 4 sets of three parallel diagonals where one is the main diagonal (skips three vertices) and two skip just one vertex. For each of these sets of three, you can choose three pairs of two. That makes 12 pairs all up.

So there are 4 + 12 = 16 pairs.