The Soviet occupation authority (SVAG)'s decision in June 1945 to allow the reestablishment of four political parties- the Communist Party (KPD), Socialist Party (SPD), the liberal democrats (LDP), Christian Democrats (CDU)- was a move that surprised many contemporaries at the time. Especially in the light of the Cold War and the later erection of single-party dictatorships within Eastern Europe, SVAG's move at first appears rather puzzling. However, there was a certain logic behind this decision. For one thing, Stalin believed that the Soviet European interests postwar would not be furthered by pursuing a strategy of revolutionary insurrection by a vanguard party, but rather through Communist Parties entering into political alliances with the left and co-opting the democratic system.

Allowing the parties to operate in their zone was both a way to claim the moral high ground in the emerging Cold War, but also signal to the rest of Europe the Soviet's willingness to participate within the democratic system. There was also a great fear that if Soviet interests were not represented in the process of reconstructing democratic governments, they would be locked out of power in Western Europe. The bitter aftermath of fascism an German occupation renewed interest inside Europe for both democratic processes and many of the figures emerging as the new postwar political leadership were individuals who were either imprisoned by the Third Reich, like Eugen Kogon and Konrad Adenauer, or emigres that lived abroad. In the latter case, communist-affiliated German emigres often suffered a stigma trying to establish themselves in the postwar order as there was a common perception that they had sat out the collective suffering of the German people and had a good war while serving a foreign power. Germans imprisoned by the regime, even if relatively briefly like Adenauer, often possessed much greater political capital than someone like KPD co-chairman Wilhelm Pieck who spent much of the post-1933 period in the USSR.

The potential independence of these political parties also was quite distressing for SVAG and their KPD allies. Although the KPD had assumed an organizational leadership in umbrella organization the National Committee for a Free Germany (NKFD) during the war, there were signs that their wartime German allies were hewing to their own agendas once the fascist threat had been eliminated. Ironically, some of the SPD members in the immediate postwar was more doctrinaire and leftist than the KPD and advocated a more wholesale restructuring of the German economy than official Marxist-Leninist German political party. Moreover, the SPD was able to operate much more freely in both Western and Eastern zones, something that the KPD was only able to do fitfully when outside of Soviet largess. Allowing the parties to operate in the Soviet zone was a means to control them. In an August 1945 meeting with his officers explained this rationale behind allowing this bloc of left to center right political parties to operate inside Germany:

I am not at all in agreement with the idea that we should maintain an identical and equal relationship towards all parties of the bloc. We should comprehensively support one party [KPD] and enable it to increase its authority, while we should keep up the others with the aim of neutralising those strata of the population who support the two liberal bourgeois parties.

This neutralization strategy recognized that a true multi-party system in the Eastern zone was inimical to SVAG's interests, but that by allowing the appearance of one while favoring the KPD would deny the "bourgeois" parties from capitalizing upon Soviet actions. The Soviets funneled resources into the KPD-affiliated press and KPD heads enjoyed access to SVAG, while other party heads did not. In a time of real material dearth inside Germany, such access was vital political capital.

But these heavy-handed tactics helped exacerbate the political cleavages between the KPD and the SPD in both zones leading to a break between the wartime alliance between left-wing parties. The failure of the Austrian Communist Party(KPÖ) to make headway in the November 1945 elections while the Austrian Social Democrats gained seats further alarmed SVAG that an independent SPD was too dangerous. Thus SVAG opted to push for a unification of both the KPD and SPD into a super-party, the Sozialistische Einheitspartei Deutschlands (SED) in late 1946. SVAG's Lieutenant Colonel Sergei Tiulpanov warned Stalin that the KPD might suffer the same fate as the KPÖ if such an amalgamation did not happen. In theory, the SED was to be a symbol of a new Germany and political culture with a KPD leadership that would properly discipline its members, both SPD and KPD. In practice, this amalgamation led to a permanent split with the SPD of the Western zone. Ironically, the SED press reserved much of the bile towards SPD politicians operating in the West during the first few years of the GDR's existence; while Adenauer may have been an closet fascist, in the SED press a figure like the SPD leader Herbert Wehner was a dangerous apostate sapping the strength of the German left. The party unification also strained those remaining SPD members inside the Eastern zone and their KPD counterparts. But the amalgamation allowed the Soviets to claim that its zone was a multi-party system with one umbrella political party, when it had become a de facto single-party state. The political powerhouse of the SED meant that it was able to reduce the CDU and LDP in the Eastern zone to pliant minority parties that represented interests like Christianity that would not easily fit into the working-class ethos espoused by the SED. The continued existence of these minority parties in the GDR was useful for staking out the claim that the GDR was font of true German culture and history, including its long history of Christian theology, in counterpoint to the decadent capitalist FRG.

So while Stalin and SVAG did allow parties to operate in the Eastern zone, it was usually on their terms and the Soviet thumb was always on the electoral scales.

Sources

Naimark, Norman M. The Russians in Germany: A History of the Soviet Zone of Occupation, 1945-1949. Cambridge, Mass: Belknap Press of Harvard University Press, 1995.

Slaveski, Filip. The Soviet Occupation of Germany: Hunger, Mass Violence and the Struggle for Peace, 1945-1947. Cambridge: Cambridge University Press, 2013.

Not really. After the end of the Second World War, Stalin promised at the Yalta and Potsdam conferences that he would allow for the nations then occupied by the Soviet forced to have free and fair elections. This was obviously a lie, as Stalin wanted to maintain the sphere of influence he had acquired during the war as to allow for a 'buffer zone' between the Western powers and the USSR (seeing as how likely World War Three looked immediately after the war ended).

Whilst non-Communist/non-Soviet-allied parties were allowed within some of the occupied nations, they didn't last for all that long due to Stalin's use of rigged elections. For example, in 1945 a fair election took place in occupied Hungary, electing a non-Communist party. However, by 1947 another election was hosted, this time being rigged, which saw, surprisingly, the Communist party win. The party then continued to imprison all opposition leaders and ban opposition parties.

Another example includes Czechoslovakia - a left wing coalition was able to seize power in 1945, with the Communists becoming the largest party by 1946 whilst, still, in a coalition. In 1948 the Communist hold on power was under threat from opposition parties, so they were quickly disposed of. The main opposition leader also happened to 'fall' out of a window (true story). With the opposition practically eradicated, a rigged election allowed the Communist party to become the only legally-allowed party in Czechoslovakia.

Examples like this can also be seen in Bulgaria, Poland and Yugoslavia - either the allowed opposition parties were 'removed', or rigged elections were held to remove them from power. So to answer your question on why Stalin allowed them, for all intents and purposes he didn't. The only reason he allowed them to exist in the first place was to give the impression that he was following what he had promised to his fellow world leaders, only to have them swiftly removed when it was 'clearly shown' (through rigged elections) that the Communists were obviously the people's choice.