Explanations of stellar parallax that I have found involve examining the apparent motion of a nearer star relative to a background of more distant stars, as the Earth moves around the sun or as the Earth rotates. We can see the apparent motion because we have telescopes. In Tycho Brahe's time, telescopes were not available, and this kind of motion couldn't be seen. The fact that stellar parallax couldn't be observed is supposed to be evidence that Tycho and others took as evidence that the Earth didn't move.
However, as I understand it, Tycho believed that the stars move together in a fixed sphere, so it seems that he would not even try to look for parallax using the near star/far star method. He would have had to look for motion of the moon or a planet relative to the stars. The moon and planets were in constant motion, however. This doesn't mean that you can't test for parallax, but doing so would be more complicated. How was this done?