I would say you might if your seated on time because there is a chance your pager might go off past 6:35 PM. Since Chef Mickey's is a very popular location they sometimes get backed up with a lot of ADR's.
I think it is going to be very tough to make it. Not only is what "CR Resort Fan 4 Life" said a very real consideration, but character meals always take longer than regular dinners. You don't say if you are traveling with children. If you are, they are going to want to see all the characters and depending on where you are in the rotation, that might factor into your time issue.
Do you really want to have to rush through to try to make it? You always could. Also, by not being there early enough to get a good spot, you might end up with a less than desirable view of the parade.
I would say get there for 6:15 the latest to ck in. Then as soon as you r done eating I would walk over cause the monorail takes at least 20 min's to get back to the MK from the Contemporay.