Preheat the oven to 375 degrees F. Melt the butter in an oven-safe 10-inch nonstick skillet over medium-high heat. Add the shallots and 1/4 teaspoon salt and cook, stirring occasionally, until the shallots are browned on the edges, about 5 minutes. Reduce the heat to medium, add the parsley and cook until the shallots are softened and slightly more browned, 5 more minutes. Whisk together the eggs, cream cheese, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl. Pour the egg mixture into the skillet, stir with a rubber spatula to combine and cook until the bottom sets, about 3 minutes. Transfer the skillet to the oven and bake until the frittata is mostly firm (it should jiggle slightly when shaken), 4 to 6 minutes more.
Remove the frittata from the oven and let cool slightly before sliding it out of the pan. Slice into 4 to 6 pieces and serve.
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