Toss the parsnip with lemon juice, olive oil, chives, salt, and pepper. Set aside to soften.
Preheat the broiler and set a rack about 6 inches from the heat.
Gently fold together the crab meat, mayonnaise, Dijon mustard, Old Bay, salt, and pepper.
Toast the bread slices lightly, until just golden.
Divide the crab mixture evenly among the bread slices, spreading to the edges. Top with shredded cheese.
Broil 2 to 3 minutes, watching closely, until the cheese is melted and just beginning to bubble.
Remove from the broiler and top each sandwich with the parsnip slaw. Serve immediately.