Hampton Court and Parliament on same day

Anything is doable if you run. Seems a bit rushed?

I think that is doable. It really depends on how long you want to spend at Hampton Court and how you manage your mid-day meal. If you arrived at 10 am when it opens and spent up to three hours, you would still be at just 1 pm and could conceivably be back in London by mid afternoon.

We found that we made it through in about that amount of time a few weeks ago. We did not spend a great deal of time in the gardens though.

I did both on my recent trip--though not on the same day. I travel solo so move fairly quickly when I want or need to, so here's my experience:

Arrived via train at Hampton Court just before 10 am opening--a Thursday, not crowded--toured everything tourable in the palace, had a relaxed lunch at the Tiltyard cafe, wandered the gardens and was on the train-40 minute ride--and back in London by 2:45. My guided Parliament tour was on a Saturday and I saw that there were tours as late as 4 pm if I recall correctly, so if you got back to London by 3 pm it's only a 5 minute tube ride from Waterloo to the Westminster station--you do need to be at Parliament at least 15 minutes early to get through their airport-style security. It could be a bit of a rush, depending on your and any companion's travel style, but I must say that touring Parliament stands out as one of the most satisfying and interesting things I've done on my trips to London, so I hope you can make it work.

As far as I know tours of Parliament are only on Saturday unless you are a UK citizen.

We spent 6.5 hours at Hampton Court.