1. In a 10- to 12-inch ovenproof frying pan (with sides at least 2 1/2 in. tall) over medium-high heat, melt butter. Add sausages and turn until browned all over, 5 to 8 minutes. Transfer to a plate.
2. Reduce heat to medium, add onion and garlic to pan, and stir occasionally until slightly browned, 5 to 8 minutes. Pour in cream and 1 1/4 cups broth; stir in salt and pepper. Add potatoes. (Liquid should just cover; if it doesn't, add more broth.) Bring to a simmer over medium-high heat, gently stirring occasionally. Lower heat and barely simmer, uncovered, until potatoes are tender when pierced, about 20 minutes (do not boil).
3. Arrange sausages on potato mixture and transfer pan to a 350° oven. Bake until cream mixture has thickened and is well browned on top and sausages are cooked through (cut to test), 40 to 45 minutes.
4. Sprinkle with parsley. Serve potatoes and sausages from pan.