Parliament alone doesn't solve the problem of fptp

The problem with any fptp voting system is that government by a single person is not necessary. If we govern ourselves in a democracy then we can have direct democracy which gives everyone an equal vote. Fptp-style elections give us the most government because they concentrate the power in the hands of one person. A way to mitigate this is to have a parliament of politicians which are sourced from constituencies. We would hope that this would reduce the problems of having only one leader. But if we are not careful to make sure the constituencies promote their politicians in a proportional way the problems of fptp re-emerge. In any voting system political parties will emerge to make it easier for people of a similar outlook to get elected. This is fine but we must recognise that each party must have a leader and this is where the problem comes from. Because each party has a leader we will get the same problem from a fptp voting system with multiple politicians as we would from an ordinary mayoral-type system.

Mayor-type systems are those which are a straight run-off between two people for a single position.

If there is only one winner in each seat then each seat will reduce into a two-party election which will be mirrored across all the seats. Having plenty of parliamentarians doesn't solve the mayoral problem of fptp. Fptp in each seat makes it fptp for the presidency.