Prepare dressing by whisking together vinegar, brown sugar, cilantro, and parsley in a bowl. Add oil in a thin stream while whisking constantly. Set aside to let flavors integrate.
Bring 3 3/4 cups water to a boil. Stir in wild rice, cover pot and let cook undisturbed for 15 minutes. Add white rice to the same pot and cook covered for another 20 minutes. Remove from heat and let sit covered for another 5 minutes to let rice absorb water, then remove lid, stir to fluff rice and let cool.
Combine onion and pineapple in a bowl and stir in cooled rice. Add dressing and stir to coat. Season with salt and pepper, as needed. Transfer to a serving bowl and garnish with cilantro sprigs.