Rub curry paste on all sides of pork. Arrange pork on a jelly-roll pan and refrigerate.
Heat oven to 425 degrees F. Bring 1 can broth to boil in a medium saucepan over medium-high heat. Add quinoa; cover and simmer until liquid is absorbed and grains are tender, 15 minutes. Remove from heat; let stand covered, 5 minutes.
Place jelly-roll pan with pork in oven. Roast pork 10 minutes. Turn and roast 10 minutes more, or until an instant-read thermometer inserted in center of each piece of pork reaches 160 degrees F. Transfer to a cutting board and let stand 5 minutes.
Meanwhile, bring kale and remaining 1 can broth to a boil in a medium saucepan over medium-high heat. Add garlic and oil. Reduce heat and simmer, stirring often, until kale is tender, 20 minutes. Transfer kale to a large serving bowl; stir in quinoa.
Thinly slice pork and divide among 4 serving plates. Serve with kale, quinoa, and yogurt. Garnish with parsley, if desired.