

In the Senate there are a bunch of rules around what can be passed without being subject to a filibuster (effectively what can be passed with >50 votes rather than >60). The parliamentarian is basically a referee who decides which bills (and parts of bills) these rules apply to, and is appointed (not a senator).
A common case is that the budget cannot be filibustered, so senators try to put stuff into the budget bill to get it through, and then the parliamentarian says “no, that’s not a budget item”.
None of these rules are legally binding, so some get upset when Democrats abide by them.
For additional context the current parliamentarian is Elizabeth MacDonough , appointed by Harry Reid (D) in 2012.
The parliamentarian does handle a lot more than just budget reconciliation (e.g. who gets to speak when), but I’m not aware of any recent controversy besides that.
I think “the parliamentarian” is sometimes used as a synecdoche (or in ignorance) for Democrats’ unwillingness to break senate procedure – or the law – in general.