The filibuster is a rule of the Senate, but the Constitution states that the Vice President is the presiding officer. Of course in practice he's rarely there and the Majority takes control, but that's their own rule. The presiding officer should be the one to interpret and enforce the Senate's rules. While Harry Reid may have to be there every day and is afraid to fight hard, the Vice President needn't be so namby-pamby.
The current Rule XXII that defines filibusters is obviously out of control. It was not designed to allow the minority to file a silent filibuster on almost everything at once. The Senate's rules were designed to allow extended debate. With insanity like Shelby's hold-for-pork, the public would be ready for some progress for a change. Especially the key independents, who are probably no fans of the Senate's sclerosis. So here's an idea of how to make it all work. It is a step shy of the nuclear option -- just a strong conventional weapon.
So let's say Vice President Biden showed up one day. He's open up debate on, say, health care reform (the House-Senate compromise, ideally with a public option). He could say, as presiding officer, that the minority is entitled to filibuster as long as they wanted. But because filibuster is about extended debate, he would no longer accept the silent filibuster. If they want to stop a vote, they have to talk. On C-SPAN. One turn at the podium per senator, so bring the Depends.
The Parliamentarian's job is to advise the presiding officer about the rules. He might disagree. He is an appointee, and the Vice President presumably has the right to override him. (Nominally non-partisan, the current one was appointed by a Republican leader.)
Debate is not about sitting silently, as imagined by the current rule, in which a filibuster simply requires a member of the minority to be present. So the chair should say that they really have to keep talking. Ten seconds without a word ends their turn, and he'll bang the gavel. A minute without ten words would do the same. So let 'em play at Strom's game, but now on TV. Eventually they will tire.
It's possible that the Rethuglicans will simply object, but in that case the majority has enough votes to call the question. So they sit this out at their own risk. While the Senate will be hung up as a couple of dozen Rethuglicans take their turn, eventually the Vice President will call the question. No cloture needed; debate is simply deemed to be over.