The Wizard government appears to take the form of a Ministry, and given the name it seems that it would be part of the Muggle UK Government. The Prime Minister is certainly aware of the Ministry, although it seems to trouble him a lot, and the PM exerts no control over it. However, the Minister of Magic is required to report to the Prime Minister, so the Magical world and the Muggle world aren’t entirely politically separate.
The Ministry seems primarily to be a beauracracy. It is headed by the Minister, and although we have glimpsed some of the political machinations, we don’t really know how somebody becomes Minister. I am inclined to think the Minister is not elected, since elections have never been mentioned, and while no particular quotes come to mind, I believe the text hints at the Minister being selected, not elected. When Fudge was replaced at the beginning of Half Blood Prince, the transition seemed rapid and, although motivated by public dissatisfaction, not the result of any poll.
It may seem odd that Wizarding society isn’t democratically run, but I don’t think it would be that surprising; it seems to be rather archaic in many of its civic matters.
My guess is that the Minister is appointed by general consesnsus of powerful and influential people, but nevertheless, perhaps to keep the peace, required to hold the public confidence - remember, Fudge had to quit when he lost goodwill.
I do not think there is a Wizard Parliament, because the Ministry seems to be described in the books as a beauracratic organization rather than a political organization. The Ministry seems to be in charge of administration while policy disputes seemed to be hacked out in an unofficial manner by wealthy, powerful and connected figures such as Lucius Malfoy. There’s no indication that Malfoy is directly involved in running the Government, but he certainly manages to affect the way the Ministry is run.
Also, International agreements seem to be more important and hold more weight in the Wizarding world. Could you imagine getting every single Muggle country to agree to the Wizarding Statute of Secrecy? Or fulfilling the requirement to keep Magical animals hidden from Muggles? Of course, the text does mention verious foreign policy difficulties the Ministry has, but these tend to be about lesser issues such as cauldron standards and the flying carpet trade.
The Wizard court, the Wizengamot, seems to differ from our court systems in that it has a very large number of “jurors,” and that the separation of powers does not exist - in Harry’s trial in Order of the Phoenix, the Ministry acted as the prosecutor as well as the judiciary, a clear conflict of interest. Again, Wizards aren’t that good with civics. Also, the right to a trial seems to be enshrined in regulation rather than statutory or constitutional law - Sirius Black was imprisoned for the murder of Lily and James without a trial.
So, in sum, British Wizards live under a beauracratic Oligarchy in a world with a reasonably high level of international co-operation. Their judiciary has the potential to be highly susceptable to corruption.