You are correct - those "parsley worms" are the larvae of swallowtail butterflies. I assume that the eggs are laid on the host plant, like most butterflies. They grow fast once they hatch, so fast it can look like they just appear one day. The caterpillars also molt several times and their appearance changes - from black and fuzzy to the striped version you saw. Once they are ready to pupate, they make a chrysalis attached to a branch or twig. I occasionally find one of these in the garden, usually when clearing or cutting back plants.

Thank you for detailed and interesting information. To photograph a close-up - touch gently and horns will appear. Beautiful.

Gusanos de perejil