This page of the guide to Final Fantasy 7 Remake has a walkthrough for the Chocobo Search side mission available in Chapter 14 - In Search of Hope.
Andrea will give you the task. After a brief conversation at the gym, Jules will ask Tifa to participate in the Pull up contest. Your job is to defeat your opponent.
This minigame is very similar to Burning Thighs. But it's is a little more difficult. However, we're about to explain everything exactly to make your task easier.
As with Burning Thighs, you have to press the buttons at the right pace and in the right order. Remember to be vigilant and not spoil the rhythm. Pay attention to the order of the keys, they change every round.
Reward: after winning the first round, you will receive Way of the Fist Vol. III. However, if you complete all pull-up tasks you must score over 43.