Heat oven to 350 degrees. Season shrimp with kosher salt and pepper, then toss with 1 tablespoon oil; cover with plastic wrap and refrigerate until needed. Season chicken thighs with salt and pepper; set aside.
Heat 2 teaspoons oil in large Dutch oven or paella pan over medium-high heat until shimmering but not smoking. Add peppers and cook, stirring occasionally, until skin begins to blister and turn spotty black, 3 to 4 minutes. Transfer peppers to small plate and set aside.
Add 1 teaspoon oil to now-empty pan; heat oil until shimmering but not smoking. Add chorizo to pot; cook, stirring frequently, until deeply browned and fat begins to render, 4 to 5 minutes. Add chicken pieces in single layer; cook, without moving pieces, until browned, about 3 minutes. Turn pieces and brown on second side. Transfer chorizo and chicken and set aside.
Add enough oil to the fat in Dutch oven to equal 2 tablespoons; heat over medium heat until shimmering. Add onion and cook, stirring frequently, until softened, about 3 minutes; Stir in tomatoes; cook until mixture begins to darken and thicken slightly, about 3 minutes. Add minced garlic, then stir in rice and cook until grains are well coated with tomato mixture, 1 to 2 minutes. Stir in chicken broth, wine, saffron, paprika, bay, and 1/2 teaspoon salt. Return chicken and chorizo to pot, increase heat to medium-high and bring to boil, uncovered, stirring occasionally. Cover pot and transfer to oven; cook until rice absorbs almost all liquid, about 15 minutes.
Meanwhile, steam mussels open in a pot with the other 1/4 cup of white wine. Reserve all the liquid. Discard any mussels that didn't open.
Remove pot from oven (close oven door to retain heat). Uncover pot; scatter shrimp over rice, and pour mussels liquor over rice. Scatter peas and peppers over top. Arrange open mussels around the edge of the pot. Cover and return to oven; cook until shrimp are opaque.
Let paella stand, covered, about 5 minutes. Discard the bay leaf, if it can be easily removed. Sprinkle with parsley and serve, passing lemon wedges and ailoi separately.