Make pickled squash: Using a handheld slicer such as a mandoline, slice squash paper-thin. Pack squash slices into a 1-qt. jar or heatproof container.
In a small saucepan, combine remaining pickled squash ingredients with 1/2 cup water and bring to a boil, stirring to dissolve sugar and salt.
Pour hot brine over squash. Cool to room temperature, then cover and chill for at least 24 hours and up to a month before serving.
Make dressing: Put vinegar, mustard, maple syrup, both oils, and salt and pepper in a pint jar and shake vigorously to emulsify.
Separate chicories into leaves and tear any very large ones in half; leave smaller ones whole. Rinse, dry, and put in a very large bowl. Add parsley leaves.
Just before serving, add 1 cup pickled squash. Toss salad lightly with salt and pepper, then drizzle in vinaigrette, starting with about 1/2 cup. Toss salad to thoroughly coat each leaf, then add more vinaigrette to taste if needed.