Place half the butter in a soup pot large enough to fit all the mussels. Turn the heat to medium and grate the onion and garlic directly over the pot to catch any juice. Add the tomato paste and stir for 2 minutes until the onion and garlic are fragrant.
Add the white wine and let bubble and reduce by half, for about 3 minutes. Add the chicken stock, saffron, and parmesan rind. Bring the mixture to a bubble and let simmer, covered, for 10-15 minutes.
While the broth is simmering, heat a separate skillet to medium-high heat with a drizzle of olive oil. Split the chorizo links in half and place in the pan - brown for 2 minutes on each side and remove from the pan. Carefully cube the chorizo then return to the pan. Toss until fully cooked and browned all over, about 3 more minutes. Remove the chorizo from the pan and reduce the heat on the skillet to medium. Use the same skillet to toast the bread in the chorizo drippings, then set aside.
When the chorizo and bread are done and the broth has simmered, add the mussels to the pot and cover tightly. Increase the heat to medium-high and steam the mussels for 10-15 minutes until most of them are open. Add the rest of the butter to the pot with the reserved chorizo, cover the pot again, and let cook for 3 minutes more until the mussels are cooked and the butter has melted into the sauce.
Before serving, remove the parmesan rind from the broth and discard it (into your dog's mouth). Remove and discard any mussels that do not open during the cooking process.
Serve the mussels topped with parsley with white wine for sipping and lots of crusty bread for dipping. Enjoy!