Combine oil, shallots, and a generous pinch of salt in a small bowl.
Quarter fennel bulb lengthwise and cut away the core. Slice quarters thinly crosswise with a mandoline or other manual vegetable slicer, or with a chef's knife.
Segment oranges: Working with one at a time, cut a slice off the bottom and top to expose flesh. Stand orange on a cutting board. With a small, sharp knife, slice off peel and white pith, following the curve of the fruit.
Cut between membranes and fruit to release segments into a bowl. Repeat with remaining oranges.
Remove core and any bitter-tasting dark green outer leaves and leaf tips from frise. Tear pale yellow center leaves into bite-size pieces to get eight small handfuls.
Combine frise, fennel, and parsley in a serving bowl.
Add orange segments to frise, leaving juice behind.
Add shallot oil, several grinds of pepper, and 1/2 tsp. salt and toss gently. Season with more salt and a squeeze of lemon juice if you like.
Serve immediately.
Make ahead: Through step 4, up to 8 hours; cover shallot oil and leave at room temperature. Put fennel in a bowl, cover with a damp towel, and chill. Cover oranges and chill. Wrap frise in a paper towel, put in a plastic bag, and chill.
