Hi Rachel,
I like to just pick the blue petals as the calyx can be prickly, however you can include that part if you wish.
For the seed oil, it would be harvested just as the seeds are ripe, just after they have turned brown. You can pick the seedheads when you see that the seeds are just ripe and pop them in a paper back then close and shake this vigorously. The seeds should come out fairly easily this way.
Enjoy!
Warmly,
Benna