Preheat the oven to 350 degrees F.
Lightly coat a 9x13-inch baking dish with cooking spray.
In a medium bowl, add the biscuit mix, its seasoning packet, and the water and stir to combine into a dough.
Spread the biscuit dough into the prepared baking dish.
Bake until the biscuit dough just starts turning golden-brown, about 10-12 minutes.
Transfer the baking dish from the oven and set it aside.
In a large skillet over medium-high heat, add the olive oil.
Add the beef to the hot oil in large chunks and cook, undisturbed, until they are well-browned on the bottom, about 3 minutes.
Stir the beef and cook, while stirring occasionally, until it is almost cooked through with some pink remaining, about 5 minutes.
Add the onions and the poblanos to the beef and cook, while stirring occasionally and crumbling the beef, until the beef is fully browned and the onions are starting to soften, about 3 minutes.
Add the taco seasoning and the salt to the beef mixture and stir to combine.
Transfer the skillet from the heat.
In a medium bowl, add the mayonnaise, the sour cream, and the smoked paprika and stir to combine.
Add 1 cup of the pepper jack cheese and 1 cup of the cheddar cheese to the mayonnaise mixture and stir to combine.
Spoon the beef mixture evenly over the biscuit dough crust in the baking pan.
Add spoonfuls of the diced tomatoes and the green chiles evenly over the beef mixture in the baking dish.
Spoon the mayonnaise mixture over the beef mixture and gently spread it out evenly.
Sprinkle the remaining pepper jack cheese and the remaining cheddar cheese evenly over the mayonnaise mixture.
Bake until the cheese is melted and starting to brown in spots and the beef mixture is bubbling at the edges, about 30 minutes.
Transfer the casserole from the oven and allow it to rest for 10 minutes.
Serve garnished with the parsley.