How this web site turned up on multiple web searches for this topic, I'm not sure. Anyway, when people are searching for information that this web site could display but hasn't, I'll try to provide the information. Although outdoor herb gardens and caterpillars are still a month or two ahead of us in the Blue Ridge Mountains, people further south may already be finding caterpillars in their herb gardens and asking whether monarch caterpillars eat herbs like fennel, dill, or parsley.
The short answer is no. Caterpillars sometimes explore plants they can't eat, or eat things that aren't part of their natural diet and don't give them proper nutrition. You might see a monarch butterfly caterpillar on a plant other than milkweed, but that's not normal and the caterpillar wouldn't eat very much of that plant. (Monarch caterpillars often choose plants they can't eat, where the creatures that eat them are less likely to look for them, to pupate and turn into butterflies.)
However, baby swallowtail butterflies, sometimes known as parsley caterpillars, devour parsley, dill, and related garden plants.
Confusion is possible because, although the caterpillars look very different, they might be described in similar terms. Both are good-sized caterpillars, eventually growing two inches long (some swallowtail larvae are bigger). Both have black, white, and yellow stripes; monarch caterpillars' stripes are narrower. Both, during the last stage before they pupate, have "horns"; monarch caterpillars' appendages are long, flexible black "whips," and swallowtail caterpillars' "horns" are a fleshy erectile organ called an osmeterium that resembles a snake's tongue and emits an odor many birds dislike.
Monarch caterpillars are slim, with straight, smooth bodies. Swallowtail caterpillars vary--there are different species that can eat dill and parsley--but most have a humpbacked shape and a stubbier build than monarch caterpillars.
Here's a primary-school-friendly web site with more information about monarch butterflies:
Here's a page with nice clear pictures of mature monarch caterpillars, showing their "horns":
Here's a chart showing what lots of different butterfly caterpillars eat:
Here's a page with (gross-out alert--highly magnified caterpillar images) clear, colorful photos of many caterpillars. You'll have to scroll a long way down for the monarchs, even further down--almost to the bottom--for the swallowtails:
Here's a stressed-out Black Swallowtail caterpillar, waving its "horns":
Here's an article about swallowtail butterflies, including nice clear photos of eggs and newly hatched swallowtail caterpillars, so you can (if you wish) pick them off your herbs before they really start devouring:
While the monarch butterfly is unique, the swallowtails are a large family, with some member species that don't eat garden plants at all. Tiger Swallowtails mostly eat the larger and tenderer leaves found at the tops of tulip poplar trees, which is why you seldom see the caterpillars. Zebra Swallowtails eat only pawpaw trees. And so on.