Heat broth, wine, saffron and bay leaves together in a large pot and keep warm.
When ready to start grilling, light a full starter chimney or even bed of charcoal. When all the charcoal is hot and covered with gray ash, set cooking rack in place close to the coals and place a 17-inch paella pan on top. Add the olive oil and use a long heat-proof spoon or spatula to move it around to coat the pan. When the olive oil is slightly smoking, add the tomatoes and sofrito sauce and cook, stirring and scraping the bottom of the pan until the sauce starts to lightly brown.
Add the piquillo peppers, paprika and 9 cups of broth mixture. Add sea salt and freshly ground black pepper to taste. Mix well, allow to boil and then sprinkle rice evenly around the pan. Stir in peas, saving some aside to sprinkle on top. Keep the pan over high heat and allow to simmer vigorously for 6 minutes. (It is important to not stir the rice after this point.)
Tuck the mussels into the rice, placing edges of mussel shells so they open facing up. Add shrimp, nestling them down into the rice a bit. If the coals are still very hot, carefully raise the pan a few inches off the heat (using wads of foil or bricks if necessary) to allow the paella to gently simmer for 15 minutes.
After 15 minutes, sprinkle on remaining peas, cover the grill and cook for 5 minutes more. If the broth cooks off and the rice isn't done enough, add more warm broth in small additions until the rice reaches the desired doneness. At the end of cooking, all the liquid should have been absorbed and the rice should be just tender.
Remove paella from grill, cover the pan with a sheet of foil and leave to rest for 10 minutes. Remove foil, discard any mussels that did not open and garnish with the chopped parsley leaves if using.
Serve hot with alioli garlic sauce, lemon wedges, crusty bread and a Spanish white or rosé wine.
Cook’s Notes for Grilling Paella:
Make sure your grill is level prior to lighting.
A covered grill which has an easily adjustable cooking rack or adjustable side vents works best. If your grill doesn’t have either, have 3-4 large, evenly sized pieces of balled up and flattened aluminum foil or 3-4 clean, dry bricks handy so you can raise the paella pan away from the heat if needed. (If using balled up foil, you will want to try them out under the paella pan to make sure it sits level prior to lighting the grill.)
It is helpful to have a flat, heatproof surface close to the grill to set the paella pan on, or someone to help you to adjust the height of the grill cooking rack, as caution should be taken when moving around the paella pan once it is full of hot liquid.
Have all your ingredients ready and near your grill prior to cooking.