What is the political party in a parliamentary democracy?
The way that this question is worded makes it difficult to know the answer. It may be that you have a clear statement in your book (or notes) that would tell you what answer your instructor wants. I suggest you look for such a statement.
I would say that the best answer for this is to say that the political party is the most important unit in a parliamentary democracy. Parliamentary democracies are built around political parties to a greater extent than presidential democracies are.
Parliamentary democracies are dominated by their legislative branches. They do not have strong executive branches that are separate from the legislative branch. For example, in the United Kingdom, the prime minister and the other ministers (the heads of various executive departments) are all members of Parliament, which is the legislative branch. This is different from how things work in the United States where no member of the legislature is also part of the executive branch.
Parties are important in a parliamentary democracy because it is the parties that determine who the head of government (often called the prime minister) and who the leaders of the executive departments will be. In parliamentary systems, voters elect their own members of parliament. Whichever party gets a majority in parliament gets to choose the prime minister and the other ministers. If no party gets a majority, parties typically have to form coalitions. The various parties in the coalition have to hammer out agreements as to who the ministers will be. In both cases, though, it is the party (or parties) that makes all the decisions.
From this, we can see that parties are the most important unit in parliamentary democracies. Parliaments are organized around parties because it is the parties that get to determine who the various ministers will be.