In East Germany and Poland, the multiple parties were part of the legacy of the Communist takeover.
In Eastern Germany, the Soviets originally banned the Christian Democratic Party and forced the Social Democratic Party to form a coalition party with the Communist party. This became the Socialist Unity Party. The Social Democratic Party was much larger, but the Communists were in full control of the Socialist Union Party. The Social Unity Party was the official Communist Party in East Germany.
With the two largest parties out of the election (The Christian Democrats banned and the Social Democrats merged into a Communist lead party), the Socialist Unity Party swept the 1946 elections and were able to proclaim Eastern Germany as a Communist state. However, the Socialist Unity Party (the official Communist party of Eastern Germany) didn’t win all seats. Instead, a few minor parties (like the People’s Party) won a few seats.
To handle this, the Soviets forced all parties to form a coalition unity government. This allowed the East German state to muzzle any possible political opposition since all parties were members of the government, there was no opposition.
In 1948, many of the former parties reformed under Communist scrutiny where trouble makers were tossed out, and in 1949, new elections proclaimed the Democratic Republic of Germany which would be ruled by the National Front of four parties. In reality, of course, these parties had little or no power and were merely suppose to represent parts of the East German public that was not represented by the Socialist Unity Party.
Like all Communist states, the government itself is secondary to the party. The Parliament only met a few days out of the year mainly to approve decisions that had already been decided. Each organization of the government had a parallel structure in the party, and membership between the two would coincide. Non-Communist party members rarely got any ministry seats, and when they did, they were secondary seats and with little or no actual power.
Why these parties remained is a good question. Lenin initially allowed some opposition parties, but constantly harassed, arrested, and executed their party leaders. Stalin stomped out any non-Communist party entirely. Maybe with the death of Stalin, there was no real need to completely eliminate these parties. Unlike the non-Communist political parties in the early Soviet Union, these parties were docile and manipulable. Since they had historical connections to the state, it was felt that they added a bit of legitimacy to the government.