Combine mayonnaise, saffron liquid and lemon juice thoroughly in a food processor; season, to taste, with salt and pepper. The mayonnaise will turn a beautiful yellow color and should be somewhat thin to combine smoothly with the potato mixture.
Cook the potatoes in a large pot of boiling salted water until fork tender, about 30 minutes. Drain potatoes well in a colander. Heat oil in a skillet until almost smoking. Saute the onion and chiles. Add the potatoes and fry until the edges are crispy and brown. Transfer to a mixing bowl. Add ham, and peas and parsley; toss lightly to combine.
Gently fold the saffron mayonnaise into the potato salad, taking care not to break up the cubes.