Preheat the oven to 180C/350F/Gas 4.
Before you start cooking, cut each squid down the length of its body and open it out to reveal the inside of the squid body. Score the inside of the squid all over using a sharp knife and cut the squid into pieces. Set aside or chill until needed.
Heat the oil in a paella pan over a medium heat. Add the onions and fry gently until softened. Add the garlic and fry for a further 2-3 minutes, or until softened. Add the chorizo and fry for 1-2 minutes, or until it starts to brown.
Push the onions, garlic and chorizo to the outside of the paella pan, then drain the soaked pearl barley well and add it to the pan, stirring well to coat it in the oil. Fry the pearl barley for 3-4 minutes, stirring well.
Add the saffron and its soaking water and stir well.
Add the padrón peppers (if using), red pepper, cherry tomatoes, artichokes (in their oil) and paprika. Stir until well combined.
Add the warm stock and bring the mixture to the boil.
Arrange the grey mullet pieces, squid, prawns and mussels on top of the paella. Cover the pan tightly with a double thickness of aluminium foil so that no moisture can escape from the pan.
Bake the paella in the oven for 20-25 minutes, or until the pearl barley is tender and the seafood is cooked through.
To serve, bring the paella pan to the table and allow guests to help themselves. Garnish each plate with a wedge of lemon.