Place a rack in middle of oven; preheat to 400°. Wrap 1 lb. baby beets (mix of red and Chioggia), trimmed, scrubbed, in a few layers of foil, place on a rimmed baking sheet, and roast until tender (a paring knife should easily slide through flesh), about 1 hour. Let cool. Using paper towels, rub skins to remove and cut beets into ¼” pieces.
Heat 8 garlic cloves, thinly sliced, one 2-oz. jar capers, drained, and ¼ cup extra-virgin olive oil in a small skillet over medium, adjusting heat as needed to keep oil at a simmer, until garlic is golden and capers have burst, 8–12 minutes. Strain mixture through a fine-mesh sieve set over a small heatproof bowl. Turn garlic and capers out onto paper towels; set oil aside.
Mix beets, 1 medium shallot, finely chopped, 2 Tbsp. finely chopped pickled cherry or other hot peppers, 1 Tbsp. mayonnaise, 1 Tbsp. distilled white vinegar, 2 tsp. coarse mustard, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and 2 Tbsp. reserved oil in a medium bowl to combine. Cover and let sit 15–45 minutes to marinate.
Just before serving, stir 1 Tbsp. finely chopped parsley and half of fried garlic and capers into beet mixture. Scoop beet tartare onto a platter and top with parsley leaves and remaining fried garlic and capers. Serve beet tartare with salted potato chips or pita chips for scooping.
Do ahead: Beets can be roasted, peeled, and chopped 2 days ahead; transfer to an airtight container. Cover and chill. Tartare (without toppings) can be made 8 hours ahead; cover and chill.