Done quietly, I think it dooms Parliament. It abdicates the moral high ground while ushering in a new King that no one has had time to work up a good hate for yet - Charles II can rally Scotland and Ireland quickly, and eventually break Fairfax and Cromwell in England. Most importantly, it isn't established in Charles II's mind yet that Parliament is perfectly capable of ruling without him and that exile sucks - 10 years in France will be required for that.
I don't think anyone would think of keeping Charles II a prisoner and dictating terms to him (like marriage to a Puritan lass, pardons for relevant parties, etc) and if they did, Charles would simply break them.
Now, murder as public spectacle - make Charles II watch as his father is fed to pigs, under a sign that says "thus for Divine Right", while people hoot and jeer, and no pretense of trial - maybe that would cow him. But my impression is that at 17, or even 20, he's simply too much under the spell of his father still.
This doesn't end the Civil Wars. There will be a few more, until either Parliament or the Crown emerge completely supreme. Probably Parliament, and by 1640 or so, England will be ready to stay a Republic indefinitely.