Adversarial politics makes no sense in a modern democracy regardless its form. The idea that a significant percentage (perhaps a majority) of your fellow citizens are enemies to you and the state is a self-defeating idea, as neoconservatism and fascism demonstrated. They cause part of the citizenry and government to be at best voiceless, useless, or dangerous to the others. This can only generate hatred and paranoia. At best it wastes resources, funding such things an anachronistic senate, and functionally useless oppositions during so-called majority government, which are by definition on a part of the elected House, sometimes a numerical minority.
Parties should play their role outside of government, as rallying points and focal places for like minds and ideas, not hereditary cults (eg. I vote X because my family always has). They definitely shouldn't be an institutionalized part of the government. The House, all the House, should be dedicated to welfare and wishes of their constituents and how they fit in to the country as a whole. Political ideology should not be the first thing considered. All members of the House, should be members of the Government, no wasted salaries or votes. If we stay with a PM who is not directly elected by the country, he or she should be elected by a majority vote of the House, not merely one party. There should be no sides in the House, just an effort to produce the best results for the country, regardless whose ideas they are.
Coalitions should be encouraged to form around ideas and issues, shifting with each, with efforts to build consensus through reasoned compromise and cooperation with fellow members. Building competition into the system beyond elections pits the House against itself, and then they try to pit citizens against each other.
You know where that leads? War against brother and neighbor and national defeat.
When only one group, or person can "win", everyone else loses.
I believe parties should only be part of the system during elections that are proportionally based. After that everyone is in the same club. Ministers should be chosen proportionally, or better yet, because of qualifications for the portfolio other than party membership. All political donations should be divided into electoral and organizational. Everyone contributes to a central electoral fund from which the runners draw equally. Then the parties could match this per candidate, no more. Party donations would be like donations to charitable organizations. Electoral donations would get the higher percentage benefit at tax time, just as they do now but no one group would benefit unfairly.
I also think the legal system would be more effective if the adversarial nature were taken out and the whole things done as an inquest for the truth, but that' probably hoping for too much. Though I hear the Nordic countries have systems that work.
Of course, all this assumes that we are capable of maturity and cooperation for the greater good. The fact that civilization has reached the modern era with continuing progress gives me hope.