Preheat the grill to medium-high heat.
In a medium bowl, mix together the shrimp, parsley, garlic, olive oil, and lemon zest. Season with salt and pepper.
Form the mixture into four patties and place on the preheated grill.
Grill for 6-8 minutes, flipping once halfway through.
Meanwhile, make the garlic-lemon aioli. In a small bowl, mix together the mayonnaise, garlic, and lemon juice. Season with salt and pepper.
Toast the hamburger buns, if desired.
Place the shrimp patties on the buns and top with the garlic-lemon aioli.