Combine the grated Parmesan, breadcrumbs, and chopped parsley in a shallow bowl; set aside.
In a second shallow bowl, whisk together the all-purpose flour, cayenne pepper, salt, and black pepper.
Crack and whisk the eggs in a third shallow bowl until smooth.
Pour vegetable oil into a 12-inch skillet to a depth of about 1/2 inch and heat over medium-high until shimmering and hot but not smoking.
Working with a few chicken strips at a time, dredge each strip first in the eggs, then in the flour mixture, back through the egg, and finally press into the Parmesan-breadcrumb mixture to coat fully.
Carefully add coated strips to the hot oil and fry until golden brown, about 2 to 3 minutes per side, turning once.
Transfer cooked strips to a paper-towel-lined plate to drain and repeat with remaining chicken, refreshing the oil if it becomes heavily browned between batches.
Serve the warm Parmesan chicken strips with ranch dressing for dipping.