Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente; drain in a colander and do not rinse.
While the pasta cooks, melt the butter in a medium saucepan over medium heat; as it melts it will foam—skim off the foam with a spoon.
Continue cooking, stirring occasionally, until the milk solids turn golden brown and give a nutty aroma and you see brown bits on the bottom, about 3–5 minutes; remove from heat immediately to prevent burning.
Place the drained pasta in a large bowl, pour the browned butter over it, then add the grated mizithra and Romano cheeses.
Toss everything together until the pasta is well coated and the cheeses melt into the butter; taste and adjust seasoning if needed.
Garnish with the minced parsley and serve immediately.