If you do something on the coat-tails of someone else, you are able to do it because of the other person's success, and not because of your own efforts.
They accused him of riding on the coat-tails of the president.
See full dictionary entry for coat-tails
on the coat-tails of someone
because of the success or popularity of another person or activity, and not because of your own efforts
He was looking for fame and glory on the coat-tails of his wife.