Get a cast iron skillet ripping hot and toss your sprouts in face side down to sear. You're looking for some good colour on them. One you've got some good char on the bottoms, roll them around to get a bit of colour on the rest of the sprouts. Add a knob of butter if needed, but keep your pan as hot as possible.
Toss your parsnips, sprouts & the crushed (not minced) garlic into a small baking pan, toss in a tablespoon or so of olive oil, mix around to coat everything, then add a few sprigs of thyme. Into the oven at 400 degrees for about 20 minutes. While cooking, be sure to pull them out and toss every 7-10 minutes to ensure nothing is sticking to the bottom and things get an even cook.
Take your cast iron skillet and add a liberal knob of butter. Get this going nice and hot, then lay your chicken breasts into the butter on high heat for about 2 minutes / side. Once you've got a nice brown on each side, turn the heat down to medium and cover with a pot lid, being sure to baste your chicken every couple minutes to keep it moist.
Once your sprouts & parsnips have been cooking for 20 minutes, turn on your broiler to high, give it all another good mix, and slide back under the broiler for 10-15 mins checking occasionally. You're looking for some nice brown edges on the parsnips and your sprouts to really cook through.
Your chicken should be in the skillet for about 10 minutes depending on the thickness of the breasts. You'll want to reach an internal temperature of 160 (by code), but I like to pull them off just a tad before, maybe 150-155. Let these rest on a cutting board covered.