The curious thing is not that they became involved (because that makes sense - pretty much everyone with a stake in Europe did, IRL); but that they joined Austria's side, which obviously shouldn't happen in this situation - Austrian emperor of considerable strength and with a lot of land in the Ottoman's sphere of influence.
Russia appears to be in the Protestant League; perhaps the Ottomans joined the Catholic League not because of anything concerning Austria or Prussia, but because they were opposing Russia.