What license do I need to make a parody/new interpretation of a song?

A parody has to make some commentary about the original work, it can't simply trade on the fame of the original as a shortcut to creativity and recognition. If a work is a parody, then that's considered a "fair use," and no license is needed. And it's a defense to a claim of infringement, so you need to be pretty sure, before you undertake using that defense, that your work is really a parody. Even being reasonably sure that your works are parodies wouldn't insulate you from being sued, but you'd want to know how strong your defense might be if that happened.

But what you seem to be describing, "new interpretations," sound more like "derivative works," which only the rightsholders have the rights to make, and which would require licenses from the publishers (owners of the musical compositions) that you plan to use. If somehow you were able to get licenses to re-write these songs, the publishing splits would be negotiated with the publishers, and generally, the bigger the hits, the bigger the share of a derivative work gets demanded by the owner of the hit song.

It seems unlikely to me that you could get these licenses for derivative works for contemporary hit songs, but an IP lawyer could help you figure out if you've got a shot at "fair use," and who to ask for licenses, if not.

Yes, if your songs were fair use, you'd be able to synch them with your own images. But again, if you're wrong, you'd be creating a second work to be sued for. And it seems very doubtful that a song that riffs off an original song without parodying it could then use images that also don't parody the original music video without the entire effort being characterized as utterly lacking in fair use.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

If your song does not comment on the original then it's NOT a parody of the original. While your song may use the original to creatively satire or parody something else or someone other than the creator of the original, that is not a basis to assert that your use of the original is fair. You're simply adapting an existing work to create another, and that requires a license. Know this: Weird Al always seeks a license -- even though his works are true parodies of either the original work or its creator. You need an entertainment attorney.