There are three things to understand. First, the Senate is incredibly permissive and senators have tremendous and nearly equal authority (it’s why being a senator is a promotion from the House). They don’t need permission to do things (this is separate from whether, as a matter of strategy or prudence, working within the conference makes sense). Second, cloture is not a requirement nor is it a permission slip to be vote. It’s merely a mechanism to end debate — it is now bizarrely filed in every single debatable motion before debate ever begins. Yes, the world’s greatest deliberative body moves to shut down deliberation before ever beginning. Third, making a motion pending is powerful. Senators can do it by simply gaining recognition on the floor (provided the senate is not post cloture and no one is there to object to setting aside the current pending business if there is any - a virtual certainty if you don’t telegraph it in advance since the floor is empty almost always). Once a motion is pending it gets a vote and either passes, fails, or is tabled. 11 or 20 are required to make it a recorded vote, depending on the time of day. The matter can be withdrawn, but only by the senator who offered it. TL;DR: THE SENATE CAN JUST DO THINGS. TRYING IS HALF THE BATTLE.

Blaming the parliamentarian is always a crutch. She is a staffer. She advises the Senate. The Senate is not bound to her advice or guidance; the Senate can choose to accept it, or not. The Chair -- not the parliamentarian -- ultimately rules, and the Senate can either confirm or appeal that ruling.