Zach Parise has not signed with an NHL team. There continues to be speculation about a number of teams still in the running. Some lists have Detroit on them and some do not. In general, the players are New Jersey, Minnesota, Pittsburgh, Detroit, and Chicago (as a dark horse). Parise himself has not ruled out any team. He said earlier today that he is still making his decision and is currently discussing things with his bride-to-be.
News on Ryan Suter is even more speculative, as the only thing that's come directly from his camp recently is that his agent, Neil Sheehy has explained that there's nothing wrong with Suter taking his sweet time to make such a big decision (and he's right). However, Reports of a plane belonging to the Red Wings taking off from Madison, WI (where Suter's farm is located) has the internet abuzz right now. There are currently no reports as to who is on that plane.
It's not much of an update because it's not a lot of information, but that's what is going on right now. We'll update more when we have more to go on.
[Update: Neil Sheehy said that decision isn't likely today on Suter. We're probably looking to see it happen on Thursday while Parise and Suter celebrate Independence Day like true Americans... independently]