Toss the strawberry slices, arugula, and Parmesan crumbles in a large bowl.
In a medium bowl, vigorously whisk together the mustard, sugar, lime juice, and orange juice, allowing time between additions to completely blend and dissolve. When adding the olive oil, be sure to add slowly in a thin stream while still whisking with the other hand. This is a perfect time to ask for an extra hand, or simply place the bowl in the center of a dish towel curled into a circle around the base to stabilize it while whisking. Taste and season with a pinch of salt, if needed, and a grind or 2 of pepper. Pour over the salad and toss.