Grease the slow cooker insert lightly with a dab of the melted butter.
Whisk soup, milk, sour cream, remaining melted butter, garlic, salt, pepper, paprika, and parsley until smooth.
Spread one-third of the potatoes in the cooker. Scatter over some onion, ham, and a handful of cheddar.
Drizzle on about one-third of the sauce. Repeat two more potato-ham-cheese layers, finishing with sauce.
Cover and cook on Low for about 5 hours, until potatoes are tender when pierced.
Stir in peas and half of the remaining cheddar. Smooth the top.
Sprinkle on the rest of the cheddar. Cover 10–15 minutes to melt.
Let rest 10 minutes. Garnish with green onions and Parmesan, then serve warm.