Answer: First off, do make sure your buckets are food grade and do not contain any residue of potentially harmful substances.
For carrots and parsnips, a lighter slightly sandy mix produces shapelier results. The mix shold also be quite rich and needs to be kept fairly moist so they grow plump and juicy. You might try shorter varieties of carrots such as "Thumbelina" or, if the buckets are quite deep any variety that appeals to you.
There are really only a few varieties of parsnip available so I think that would be a matter of personal preference and possibly trial and error. Since they will be in a container and are often harvested after cold weather sets in, you may want to insulate the outer edge of the container to keep them from freezing prematurely.
Potatoes can certianly be grown in a container but I am not sure thatyour buckets will be big enough. One technique is to take a large trash can and punch some drain holes in the bottom, then place about a foot or so of soil in the bottom, then plant the eyes into that. As they grow, cover them with straw or old chopped leaves and water well. Keep doing this all summer until you have a nice crop of potatoes. Since they are very clean and easily accessible you can even steal some to eat as needed through the season.
For more detailed information about growing vegetables in containers and more ideas about other types of plants to try in your buckets you might enjoy "Container Gardening for Dummies" by Bill Marken, ISBN 0-7645-5057-8. The book also includes a more detailed discussion about soil mixes and container plant maintenance.
Have fun with your project!
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