Rinse and cut the vegetables and arrange on a platter.
For the dip, squeeze the lemon juice into a small bowl. Add the garlic and let sit for a few minutes (this takes a bit of the edge off the raw garlic).
In food processor, pulse together the lemon garlic mixture, tahini, and parsley until the consistency of a thick paste. With the motor running, add the olive oil in a steady stream. If you’d like a thinner dip, add a few tablespoons of water. Season with salt and pepper to taste.
NOTE: The dip makes about 1 cup. I didn’t write the exact amount for each vegetable because it depends on how many guests you have over, and what your prerence for each vegetable is!
MAKE AHEAD: You can prep the crudites in advance if you like: just place the cut veggies in ziploc bags or in tupperwares covered with moist paper towels in the refrigerator until ready to use. You can make the dip a day ahead and store in a covered container in the refrigerator – it will actually taste better the second day.