Halve the ripe avocado, remove the pulp with a spoon and puree with lemon juice and oil. Season to taste with salt, pepper and cayenne pepper.
Peel the parsley roots and cut or slice into very thin slices (approx. 2 - 3 mm). Heat the butter in a pan and fry the root slices over medium heat, turning several times.
When the chips have a nice color, simply drain on kitchen paper, season with salt and serve with the dip while still warm.