I know that Western countries have had amusement parks for about a century, however I have never heard about, not been able to find out if they did in the USSR. Or in any other communist republic (Cuba GDR, etc)
They most certainly did have them. One of the spookier parts of Chernobyl today is the Pripyat Amusement Park which was scheduled to open before the nuclear disaster intervened. /u/Minardi-Man 's answer covers a lot of the basics of amusement parks especially as the state authorities, especially during the Stalin period, tried to thread the needle between pure escapism and making leisure educational and uplifting.