Hi, My name is XXXXX XXXXX X am your Etiquette Expert.

After the newlywed's first dance many couples choose to honor their parents, grandparents,
and wedding party with special spotlight dances. Traditionally these special dances are
kicked off with the bride dancing with her father, followed by the groom with his mother and
then a dance with the bride & groom, parents, grandparents, and the entire wedding party
announced one by one. Of course these dances are all optional and can be adjusted according
to personal preference. After the completion of these special dances, the party begins and guests are invited to join the wedding party on the
dance floor.

With respect to your specific question - If you know the other set of parents then this would be natural to dance together. However, if you do not know them very well this would be optional. In the situation where you know the other parents well it really doesn't matter who asks who.

I hope this clarifies your question. If you have any related follow up questions I would be pleased to answer them for you.