Are the party and the government one entity or two? It makes all the difference whether "know" or "knows" is correct.
Yes, in the past communist regime, the ruling entities were the party and the government. The most powerful man was not the prime minister nor the president but the party chairman (who may have also been the prime minister or the president at the same time).
I think the question was whether singluar or plural should be used, and plural should be used in this case, these are truly two entities.
Strana a vláda vědí všechno.-Party and government know everything?