The full abstraction for PCF was solved in the mid 1990s by Abramsky/Jagadeesan/Malacaria [1] Hyland/Ong [2] and Nickau [3]. All three appeared simultaneously. This was a paradigm shift, because all three used used interactive rather than functional models of computation. (There was also later work on domain theoretic full abstraction, e.g. OHearn and Riecke [4], but I can't recall details. Maybe Streicher's work was in this direction?) The beauty of interative models like games is that they can naturally encode more complex behaviour, including parallelism.
[1] S. Abramsky, R. Jagadeesan, P. Malacaria, Full Abstraction for PCF.
[2] J.M. E. Hyland, C.-H. L. Ong, On Full Abstraction for PCF: I, II, and III.
[3] H. Nickau, Hereditarily sequential functionals.
[4] P. O'Hearn, J. G. Riecke, Kripke logical relations and PCF.