Another factor in a parliamentary system is the monarchy, which consumes huge amounts of resources, all paid for by the ordinary citizen, while performing duties which could adequately be discharged by ordinary government officials such as the Secretary of State.
A few points on the monarchy, what it does, and what people in the UK think of it.
I'm afraid I don't have proper statistics on this, but I have heard that the Monarchy costs around £100m per year. However, the monarch also owns a large amount of land (the "Crown Estates"), all revenue from which goes to the treasury. This is (I believe) larger than the cost (I think it comes to about £150m).
Supporters of the monarchy would argue that this means the monarchy is a source of revenue (not counting tourism).
On the other hand, anti-monarchists (especially the more socialist ones) insist that all that land was originally stolen from the people by the monarchy/aristocracy, and if the monarchy was abolished, it would be confiscated by the state, and so the revenue argument is irrelevent.
The queen is the "head" of various national institutions, including the armed forces and the Church of England. Some argue that it is unacceptable to have one person in all these roles,
especially when that person is an unelected and unremovable hereditary monarch. Others argue that by having a virtually-powerless "figurehead" in these roles, it prevents them being taken over by power-hungry PM (or his cronies), who would have the power to control them.
The queen has very little real power, but there are a few "royal perogatives" that in theory, give the queen unchallengable power to declare wars (among other things). In practice though, the constitution (we do have one, sort of) transfers these powers to the PM.
Anti-monarchists say this is unacceptable, for obvious reasons.
I don't know what the official monarchist counter argument is.
I would say, though, that I don't see why the monarchy could not be maintained, but stripped of these powers. Or that the monarchy could be abolished, but the powers maintained by the PM.
Another issue with the monarchy is that the monarch is supposed to be the fighure-head and representative of the nation.
Anti-monarchists argue that an hereditary aristocrat, with large quantities of unerned (inherited or tax-funded) wealth cannot possibly be representative of the people, and is not a good thing to have as a representative of the nation or state.
Pro-monarchists argue that a non-elected figure-head is less devisive than an elected one. The proportion of Britons who hate the queen is much less than the proportion who hate Blair (or whoever else is PM at the time), or (I expect) the proportion of Americans who hate POTOS. This will be partly because lots of people will have not been worked up in a camagne to elect a different queen, but lost, and partly because the queen has little power and less policy-making role, and so cannot be held responsible for unpopular government policies. The same applies to international relations: the queen can (may) be seen as a representative of Britain and the British people, rather than as a representative of the government.
And that's probably far more than you need to know, because the queen really have very little to do with running the UK.
In terms of anacronisms / hard-to-justify features of UK politics, you might want to look at the House of Lords (which has recently been "modernised" - ie all the aristocratic, mostly Tory-supporting Heriditary Peers have been kicked out, only to be replaced by New-Labour-supporting government apointees).