Clean mushrooms with a damp cloth and slice evenly; trim stems if needed.
Melt butter in a large pot over medium heat until foaming but not browning.
Add chopped onion and a pinch of salt; sauté until translucent, about 5 minutes.
Stir in minced garlic and grated carrot; cook 1–2 minutes until fragrant.
Sprinkle flour over the vegetables and stir constantly to form a light roux, cooking 1–2 minutes.
Remove pot from heat briefly and stir in sweet paprika (and smoked paprika if using) to bloom the flavor without burning.
Return to heat and slowly whisk in stock and water until smooth; add bay leaf and caraway if using.
Add sliced mushrooms, bring to a gentle simmer, and cook 10–12 minutes until mushrooms are tender.
In a small bowl, whisk sour cream with lemon juice; temper with 1–2 ladles of hot soup to avoid curdling.
Slowly stir tempered sour cream into the pot off the heat, then warm gently (do not boil) to combine and thicken.
Taste and season with salt and pepper; adjust acidity with extra lemon juice or vinegar if needed. Remove bay leaf and sprinkle chopped parsley before serving.