Israel is vaccinating Palestinians who are Israeli citizens. It actually runs special promotion campaigns (including the PM visiting Nazareth) because some in Arab population are reluctant to take it.
It does not vaccinate Palestinians in the Palestinian controlled areas as it has no authority to do so. Furthermore, Abbas and the PA leadership make a point of not allowing their citizens access to Israeli healthcare (though they did make sure to get vaccinated themselves).
Some time ago my wife was involved in an effort to provide life saving medical care to a Palestinian child in an Israeli hospital. The hospital waved all costs and the treating surgeon was in daily touch with her. The PA refused to allow the child entrance into Israel to receive that care. This wasn't an isolated case, it's PA policy.
And sure, as a fellow Jew I'd like to believe we're better than everyone else, but hey, we're not. I guess maybe better than some.
The complexity of the Palestinian/Israeli conflict can't be reduced to these simplistic terms. Anyone can take some specific incident or piece of data out of context and make their case around it, and all sides take every opportunity to do so while at the same time refusing to accept any responsibility to how they got to where things stand today.
I do hope that once Israel has finished vaccinating its population it will extend the additional vaccines to the PA and to Gaza, I think it's in their own interest and it's the right thing to do.
[0]https://en.wikipedia.org/wiki/Palestinian_Authority_Martyrs_...