Soak the eel in brine overnight. Drain, then soak the brined eel in fresh water for one hour.
In a pot, sweat the onions in olive oil. Add salt, then vinegar, water, garlic, rosemary and thyme. Simmer for 15 minutes. Drain and set aside.
Fill another pot with water and bring to boil. Put the eel in it, cover and cook for 15 minutes. Drain and cut the eel in pieces about 15 cm long. Set aside.
In a large sterilized glass container, put the pieces of eel, the eggs, the onions, the garlic, the rosemary and thyme sprigs, and the parsley. Fill the jar with olive oil and seal hermetically.