The Jim Crow South is a relatively bad analogy. I prefer to think of the occupied territories as chunks of land that Israel is sitting on mostly for defensive reasons. They were used as bases of attack in 1967, which forced Israel into a war. Israel occupied the land and has never annexed it, so it is in some kind of limbo. If all of this banter about “God given land” and “necessary for our defense,” was 100% believed, Israel would have annexed the land a long time ago. The fact that they haven’t says something about their views on future Palestinian statehood.
Increasingly, over the past 10 years of Palestinian nationalism, the occupied territories, especially the sections which are controlled by the PA, have come to seem like another country, complete with checkpoints and border crossings.
So, a better analogy may be like Mexicans coming over to work in the farms in the Rio Grande Valley. They are not entitled the US vote because they are not American. They are cheap labor from an exogenous source. Sad, but true.
To answer your OP fully, during times of peace there is quite a lot of interaction. Many Palestinians work for Israelis, as Israeli industry (like American industry) is always looking for a source of cheap labor. This says nothing about what Israelis think of Palestinians – it says more about the education level and infrastructure and domestic economy of the PA. It is the PA’s job, not Israel’s (at least since Oslo) to provide education, infrastructure, economic support, etc, for their people.
In Israel proper, there are Israeli Arabs who constitute 12% of the population. While I can’t sit here and claim that they are sitting at the same socioeconomic level as the Jewish Israelis, I will say that they are afforded the same education, health care, governmental rights and protection as any citizen of Israel. There are many in all levels of Israeli society, including parliamentary representation. It is true that they tend to be poorer than Israeli average, but IIRC Israeli Arabs are better off in most ways (freedoms, economic status, education, health care, religious tolerance) than their neighbors in Arab countries.