Rinse beets and remove the top and bottom, so it can sit flat on either side. Place on a piece of heavy duty aluminum foil and drizzle olive oil on top. Rub it in to fully coat each beet. Sprinkle with salt. Wrap in foil, place on a cookie sheet and roast in a 350 degree oven for 1 hour. Depending on the size of your beets, it could take longer. When tender, remove beets from oven and set aside to cool. When cool, gently pull or rub the skins off the beets. Cut beets into bite-sized cubes and set aside. You can do this in advance and store the beets until you’re ready to make the salad.
Combine red onion and vinegar in a bowl and soak for 10 minutes. Drain and discard vinegar, returning onions to the bowl. This is done to mute the strong onion taste.
In a small bowl, mix together the lemon juice, olive oil and balsamic vinegar. Set aside.
Add bulgur, feta, parsley and dressing to the onions and stir together. Salt and pepper to taste. Add the beets and gently toss. The beets will color the feta, so be mindful of this from a presentation aspect, but it definitely does not affect the taste. Serve immediately.