This links to another way in which the superpowers were increasingly involved in the conflict in the Middle East, which is through the support of the Arabs and the Israelis during the conflict in the 1960s. It is estimated that the US donated $2bn to Israel and the USSR donated $2.5bm to the Arab states during the Six Day War of 1967, and this is not excluding the tanks and MiG fighter planes loaned out to Egypt by the USSR before the war. After the Six Day War, Israel became the primary ally for the Americans in the Middle East, and was the prime recipient of armaments and economic funding. This shows the increasing influence of the superpowers in the Middle East, and the increasing use of the Middle East in the proxy war between the US and the USSR.
This links to another way in which the superpowers were involved in the conflict in the Middle East, but this time, to partake in peace talks to end the Yom Kippur War. American support of Israel had been heightened during the Yom Kippur War, and Arab suspicions of the West were rising. The OPEC states increased the price of oil by 70%, cut production and stopped delivery of oil to the US, resulting in the Oil Crisis of 1973, which was very damaging to the US economy. In addition, there was fear that the Yom Kippur war could expand into a larger scale conflict between the US and USSR themselves, and so, on the 22nd of October 1973, peace talks took place between the USSR and the US to try and resolve the issue. This shows how the superpowers became so involved in the conflict, in the way that they began to resolve the conflict.
Overall, the ways in which the involvement of the superpowers in the Arab-Israeli conflict were through the initial support of Israel through private donations from the US, linking to how the USSR began to support the opposite side - the Arabs, to gain a Cold War foothold in the Middle East, linking to how the superpowers became increasingly involved in the conflict, to the point where the Middle East became a site for the proxy wars of the Cold War, linking to how the focus shifted to preventing further conflict in the Middle East, to prevent the war from expanding further.