I have been thinking about your question for a while now, because I can answer it only partly. But since its still unanswered, Ill give you my thoughts.

My specialty lies in the Soviet side, but it is very difficult to me to get an estimate of how influential the US military-industrial complex was because I am simply not that familiar with US politics.

To be clear, the Soviet military industrial complex was incredibly large and influential, but there were limits. This began already in 1928, when the Soviet Union moved to implement its first Five Year Plan, in which Soviet military commanders like Kliment Voroshilov had been pushing for more industrial investments for military projects. The Communist Party agreed to proceed with planning as if "war threatened" the USSR.

During the Cold War, the military-industrial complex dominated the Communist Party's Central Committee, thus having a decisive influence in the distribution of budgets and investments. Nikita Khrushchev however undercut this dominance by reducing the Red Army by 1 million troops and decreasing its budget, arguing that 300 ICBM's simply replaced the need for massive conventional forces. Ironically, when in 1960 a US spy plane was shot down above Soviet territory, the military industrial complex began to gnaw at Khrushchev's reputation. The incident revealed the recklessness of Khrushchev's defence policies.

Khrushchev didnt last long in power, and the ones who ousted him were themselves products of the military industrial complex - chief among them Leonid Brezhnev.

Brezhnev firmly believed that in order to pressure the West into a sort of settlement, and to make it recognize and accept the position of the Warsaw Pact in global affairs, the USSR needed military strength. Khrushchev had entered the Cuban Missile Crisis with a bluff - he had not a single missile capable of reaching the US from Soviet territory. Brezhnev operated the other way around: he refrained from confrontational diplomacy, but he massively increased investment in defence industry. By 1970 the USSR was about to overtake the US in numbers of nuclear missiles, and to Moscow this was reflected by the fact that America was willing to sign several treaties with the USSR regarding arms limitations between 1969 and 1976.

In Europe too, the 1975 Helsinki Agreements signalled European recognition of the Soviet bloc and the establishment of broader diplomatic and economic relations. For this reason, the USSR continued to staunchly update and expand its arsenal from 1976 onwards, with the SS20 missiles. The push came from the general staff of the Soviet military, who after 1991 admitted they expected these missiles would never be used and were rather pointless. But as long as they pressed for the need of missiles, they could justify and hold on to their enormous budgets.

The deployments carried on between 1979 and 1984, despite vehement NATO protests and NATO eventually doing what the USSR feared most: deploying new Pershing and Cruise missiles. The era of extreme tensions and arms race between 1979 and 1984 was back then referred to as the New or Second Cold War. It wasn't until 1986 that Gorbachev, who was known for his desire to break with Soviet traditions, broke the deadlock by offering to withdraw missiles and start negotiations that led to the 1987 INF treaty.

In the political sense, the Soviet military industrial complex was represented in the Kremlin under Brezhnev by Dmitry Ustinov, a Politburo figure and military hawk who also pushed for the Soviet invasion of Afghanistan in 1979. He died in 1984, when most of the old guard was dropping dead (Brezhnev, Suslov, Andropov, Chernenko).

Other powerful lobbies inside the Communist Party were the provincial secretaries and the International department (overseeing diplomacy, the KGB, CPSU propaganda and foreign Communist parties). The latter was represented in the Kremlin largely by figures such as Andropov, Gromyko, and Gorbachev.

Recommended literature:

Cambridge History of Russia Vol. III Vladislav Zubok, A Failed Empire Taubman, "Gorbachev" Taubman, "Khrushchev" Sheila Fitzpatrick, "On Stalin's Team" Cambridge History of the Cold War Vols. I, II, III Rober Service, A History of 20th Century Russia Thomas Crump, Brezhnev and the Decline of the Soviet Union Mark Sandle, Brezhnev Reconsidered