By the 1960s, the Israeli-Arab conflict (because, to be frank, "Palestine" was not a state or a declared state at this point) was a subset of the larger Cold War. However, this is not to say that it was a clearly delineated line.

It was during the 1960s that the line began to separate the two "sides". The US, which had worked with Israel, was not Israel's main ally at this point: instead that was France, through which Israel obtained many of the planes that helped ensure its success in the 1967 War in six days. At the same time, France was not always a straightforward ally in the 60s; with the rise of De Gaulle, Israel effectively lost its ally in France, and the US filled the gap. This happened for a variety of reasons, not least because the Arab states around Israel began to align more with the Soviet Union. This effort was largely led by Nasser, the leader of Egypt. Nasser, as I explained in this answer, had begun to shift towards the Soviets in the 50s, when Eisenhower was President. Still, the US harbored some level of hopes of neutrality in the 1960s, which largely went out the window as Nasser began to sign more and more deals with the Soviet Union for weapons. Nasser, as a very popular figure and leader throughout the Arab world, was hard for any other Arab leader to oppose; indeed, by some accounts he forced Jordan to join the defense pact on May 30, 1967, which was the last straw that led Israel to launch its preemptive strike on June 5, 1967. There had been fights between the US and Egypt over more aid, conflicts over Egypt's adventures into Yemen's civil war, and so the drift apart accelerated. The US was already at odds often with Syria, a more extreme Arab state when it came to Israel and Middle Eastern politics, and Jordan was perhaps the friendliest but unable to oppose Nasser's pull, particularly given Nasser's ability and threat of riling up Palestinians in Jordan, who comprised the majority of Jordan's population and could topple the Hashemite regime. Indeed, Jordan's King Abdullah had been assassinated by a Palestinian in 1951, allegedly by an extremist who may have opposed potential peace talks with Israel that were being explored. This had an effect, some argue, on King Hussein, who realized he had to cater to his large Palestinian population, who admired Nasser for his vitriolic statements and threats against Israel.

As the ties began to sever between the US and Egypt + the Arab world, they strengthened as mentioned between the Arab world and the Soviets. The Soviets began to supply arms and advisors to the Egyptians and Syrians, and frequently joined them in diplomatic statements at the UN against Israel. Interestingly, while the US had initially had good relations with the Arab world and the Israelis, the Soviets had essentially lost major ties with Israel by the early 1950s, amid rising antisemitism and a drifting apart motivated also by Israel's socialist founders wanting to align more with the West rather than adopt full communist policies.

The Six Day War, as I mentioned, solidified the alliances. The war, in fact, began in part because of a Soviet intelligence report provided to the Egyptians that alleged that Israel was preparing to attack Syria (falsely) and that troops were being massed on the border. This led to a spiraling crisis that eventually led to war. Following the war, the Soviets' weapons had proven ineffective and the US was willing to provide more weapons to Israel, which had shown a level of military strength they did not expect.

During this period, the Palestinian groups at issue had not yet declared any state (that would wait until 1988), though they obviously viewed the area of Israel as Palestine in total. The groups were, however, largely subsidiaries operating with the consent (and often at the pleasure of) the larger Arab states. Palestinians during this period were essentially "used" by Arab states as proxy forces. Thus, while the Arab states were helped in the proxy war against Israel by the Soviets, and Israel was helped in the proxy war against the Arab states by the US, the Palestinian militias were subsidiary proxies of the Arab states. The groups themselves wanted war in the lead-up to the 1967 war, though some other Arab states may not have wanted war; they believed that the Arab world would win any such war and often wished to help provoke it. Following the war, the Soviets pushed propaganda once more about a global Jewish conspiracy, which was what they'd also initially pushed leading to splits with Israel, and in doing so helped cement their alliances with the Arab states. Some scholars even believe that the Soviets provided the false intelligence report in order to help start the war, because they wanted to force the Arab states to buy more weapons, though I doubt they expected it to go so disastrously if they did (and I have skepticism about the concept). Still, they did become more important to the Arab world after the war, though eventually they began to fall out with Egypt somewhat in the 70s, and eventually Egypt shifted to the US camp (as did Jordan, more quietly) by the late 1970s and early 1980s.

Hope that helps a bit, it's hard to write so broadly on this subject