You have to install separately on each LPAR.
Three LPARs means you have three separate system images in one hardware box.
One major point about having separate images is to allow you to make changes to one system image without disturbing the others. You apply PTFs (or other changes) to one LPAR, and the other LPARs won’t change. This lets you test the changes before applying them everywhere.
Often, the fact that PTFs <b>don’t</b> apply outside of an LPAR is the reason additional LPARs were configured in the first place. If you want PTFs to apply equally, then don’t separate into multiple LPARs.
It’s quite possible that PTFs for one LPAR are not appropriate to other LPARs. Different LPARs can be running different releases of the operating system and other products or applications.