Your question sounds like you're under the impression that the communist party was a singular, uniform entity globally. It was not. Communist parties were local, with individual parties existing for each state, occasionally multiple for individual nationalities or regions inside a state.

Before I talk about democratic successes of communism, I'm going to address the ideological approach to attaining power democratically. Communism in its most prominent and well known form is based on the economic and social theories phrased by Karl Marx, in most cases with adjustments. The most widespread form of communism that was practiced at the height of communist history is Marxism-Leninism and theories derived from it, although this was only ideological which bore little consequence in actual policy. The ideologically proposed sequence of events in communist theory was the following: the communist party, in some way, takes power (more on this later), and establishes socialism. Socialism can be summed up as collective ownership of the means of production: that is, any entity that produces economic value should be owned by society at large or the people working within or on that entity (be it a factory or a farm), and not those who claim ownership of these means without contributing their labor. It was proposed that as socialism would progress, it would lead to a social transformation that would eventually allow for the establishment of communism, a society without class, currency, and completely reliant on the cooperation of workers.

The most prominent division between Marxist approaches to politics was how the party which sought to establish socialism would take power. Two wings arose from this division: democratic socialists (called social democrats, before that became its own, less radical ideology) and revolutionary socialists. The most prominent communist states in the world, that is the USSR and the People's Republic of China, both came about through violent means, that is, civil wars started by revolutionary attempts or activities. This meant that the biggest socialist entities in the world, the actual states whose entire populations were governed by, and whose foreign policy was dedicated to socialist principles, were ideologically predisposed to favor revolution rather than democratic methodology. Also, it is notable that after the Soviet Union became a cemented power in European and later global politics, an extremely notable split took place between revolutionary socialists (usually calling themselves communists, which they technically were) and democratic socialists (usually calling themselves social democrats). This is especially visible in early 1930s Germany, where the two leftist parties had propaganda directly against each other. Communists, regardless of their revolutionary intentions and goals, still participated in elections, it just wasn't their preferred form of taking power.

One last point before we discuss the historical examples of communist electoral successes. Why were some communist ideologies in preference of violent revolution? The main reason was that they believed the type of democracy that existed under constitutional monarchies and capitalism was bourgeois, that it outright favored the preexisting ruling class, and that it was not a viable path to the working class taking power. They also had an ideological belief that supported revolution: complete revision of political and social order, to erase the past completely and start anew.

Now then, the occasions where communists took, or attempted and almost succeeded in taking power democratically. Before we move on to the most prominent, glaring, and often cited example, I'll talk about a lesser known democratic or semi-democratic attempts at socialism. Both of the examples I'll talk about will conveniently be about the same country, Hungary. Hungarian communism was first established in 1919. As Austria Hungary collapsed, an independent Hungarian state was created through peaceful revolution, democratic elections were held... and the communists gained technically no power, as they had no place in the government. In massive oversimplification, the following happened: the armistice between the newly created Hungarian state and the entente had yet to be formalized. The French had proposed the creation of a demilitarized zone and attempted to mediate a peace between Hungary and the Czechoslovak, Romanian, and Yugoslav armies that had occupied territories of prewar Hungary. In a misguided attempt to reach a democratic consensus, the government compromised with the communists, who promptly declared a soviet republic, seeing as they now had power in the government. This was a peaceful coup d'etat by all possible approaches and yet it came about through a democratic process and although it was claimed to be a revolution, it certainly wasn't reminiscent of one.

The second Hungarian example I want to give is the communist takeover in Hungary after the second World War. Fair elections were held in 1945, with the social democratic and communist parties each taking 18% of the vote. Nevertheless, they were assigned positions in the government, primarily as a result of soviet pressure, but also due to the fact that the independent freeholders' party, who had secured an absolute majority in the elections, did not want to take on the responsibility of rebuilding and reorganizing the country alone. Negotiations ended with the communists being given the ministry of internal affairs, one of two deputy prime Minister posts, and other less consequential positions. In the next 3 years, the communists in the Hungarian government, embodied mainly in the then General secretary Mátyás Rákosi, applied the so called salami tactic, a form of divide and conquer where individuals or political cliques were cut off by temporary political alliances, only for the allies to be the next ones cut off. A notoriously fraudulent election was held in 1948, and the communist and social-democratic parties were merged, and after the electoral victory, the People's Republic was proclaimed. This is, again, not an example of genuine public vote giving power to the communists, but rather an instance where violent revolution was cast aside and communist power was attained within a democratic framework.

Now, to move on to semi-legitimate examples, where 'communists' were put into power democratically. First, Guatemala. In 1951, Jacobo Árbéns was elected president. Although not a communist himself, his government did have communist members and was influenced by socialist ideology. Guatemala, like most of Central America in the 1900s, was massively within the grip of American economic influence, especially from the United Fruit Company, which essentially governed several countries in the region through puppet presidents. Árbéns had begun to implement a program of land reform which favored the peasants of the country and was against US economic interest. In 1954, he was overthrown by a CIA backed coup, staged against him both because he was believed to be a communist sympathizer, and because the UNFC did not want to suffer the consequences of losing control over one of its puppet states.

And now, finally, we get to the most frequently cited, and most known example of democratically elected communists. Salvador Allende was a Chilean democratic socialist, and was elected president of Chile in 1970. He was a self-proclaimed Marxist and relatively had relatively close relations to Cuba, the most prominent communist state in the Americas. During a CIA backed military coup in 1973, Allende is alleged to have killed himself with an AK-47 he was gifted by Fidel Castro.

So to sum up, yes, there were occasions on which communists and their sympathizers took power democratically, though it never resulted in the establishment of a genuinely socialist state. Not all communists opposed taking power in democratic manners but in a free democracy, the authority and coercion required to attain stare socialism was near impossible, while violent revolutions created both the justification and normalization required for the coercive methods that were characteristic of the social engineering and forced progress characteristic of cold war socialism.