Heat a dry skillet over medium-high heat. Place serrano chiles and garlic in the hot skillet. Cook until blistered and browned on the outside and slightly softened, 5 to 7 minutes. Roast poblano over an open flame until blackened all over; wrap up in a paper towel to let steam.
Place serranos and garlic in a bowl; let cool until safe to handle. Peel the garlic and remove stems and seeds from serranos and poblano.
Place cilantro and parsley in a blender. Add peppers, garlic, salt, and olive oil. Pulse on and off, scraping down the sides if necessary, until puree is fairly smooth and slightly coarse. Store pesto in a bowl until ready to use.
Preheat the oven to 450 degrees F (230 degrees C).
Halve and quarter chayote lengthwise. Remove seeds and cut into 1-inch pieces. Place on a lined baking sheet. Toss with olive oil and salt.
Bake in the preheated oven until chayote is just barely fork-tender and lightly browned, about 30 minutes. Transfer to a bowl and add lime juice, 2 to 3 heaping spoonfuls of pesto, and goat cheese. Transfer salad to a serving dish and garnish with pumpkin seeds.
