The parliamentary system is a form of government, proportional representation is a voting system. Most countries that have proportional representation also have some form of parliamentary system, but it is not necessary. England and Canada have parliamentary systems but they do not have proportional representation. There are many systems that can be used. Here are a few examples.
All parties provide lists of candidates which are displayed on the ballot.
The voter votes for a party rather than an individual candidate. Some party list systems allow the voter to also give his preferences as to individual candidates, but those candidates must be in the same party that he is voting for. The seats up for grabs are then allotted to the different parties according to the percentage of the vote they received. It is a very straight forward system and ensures proportional representation but it is not very likely that Americans will like the idea of voting for a party rather than an individual candidate. This system is used in Germany and a version of it has been proposed for use in Senate elections in this country.