In the case of "representative democracy", the common sense makes you think that you elect congressmen or speakers to represent your wishes, and in the "parliamentary democracy" you do almost the same; so what is the difference?
Many people use them interchangeably. As far as I'm aware, there's a subtle distinction if one goes by formal definition (which is not always in line with common usage):
A parliamentary system is a system of democratic government in which the ministers of the Executive Branch derive their legitimacy from and are accountable to a Legislature or parliament; the Executive and Legislative branches are interconnected. It is a political system in which the supreme power lies in a body of citizens who can elect people to represent them. (Wiki, emphasis mine). A classical example is of course UK.
A representative democracy is merely a system where a small subset of people (representatives) are granted power to legislate/rule on the behalf of the parts of the populace they represent.
Every parliamentary system is a representative one (since the Parliament is a form of representation, by definition).
However, not every representative system mast be a parliamentary one. For example, presidential system - like USA - have executive branch is 100% independent of legislature.
The term representative democracy is principally used to distinguish it from participatory (or direct) democracy, in which citizens directly debate and vote on legislation, policy and the appointment of citizens to execute those policies. With that in mind, a Parliamentary democracy is just an individual model of how a representative democracy might work, in that representatives are elected who in turn elect an executive (the Government). A different (and still representative) model might just involve selecting legislators by random ballot.
It is also worth pointing out that a Parliamentary democracy need not be purely representative, as events like referenda or plebiscites are examples of more direct democracy and can happily exist alongside an elected Parliament. Furthermore, the concept of "representative democracy" can be used in a normative way to discuss the way in which a legislature or executive is or isn't demographically representative of the people it supposedly represents. For example, Westminster currently has more male MPs than there have ever been female MPs, so it is a "Representative Democracy" but not a "representative" democracy.
In most cases they are interchangeable concepts.
Perhaps one could come up with corner cases where they are not the same, depending on how exactly one defines the terms, but those differences would not be particularly useful and it would be hard to agree on them.
Parliamentary democracy you vote for legislatures who vote for the leader. Representative democracy you vote for the legislatures and the leader.
In the Legislature, the representatives and senators are not Ministers in the government headed by the Executive whereas in a Parliamentary system the members of parliament serve also as ministers in the government. Moreover, in a legislative democracy, the people elect the President for the purpose of enforcing laws and separately elect the legislators for the purpose of making the laws. While in a Parliamentary system, the members of parliament are the ones that elect the prime ministers and not the people directly.