I have tried growing carrots now and then, but they were always twisting around all of the rocks and squeezing through the clay, and I basically got a bad crop of strong flavored, tough carrots that were good for nothing. Would a round variety like "Paris Market" be better for my type of soil?
The direct answer to the question you asked is a hesitant "yes". I hesitate because you may have other issues that prevent you from getting a good crop. But a short variety can help you get a usable crop if you don't want to go to a lot of effort improving your soil.
In my opinion, improving the soil is a worthwhile effort, so here's my recipe for getting a good carrot crop on marginal soil:
The soil here is very rocky, acidic, and low in organic matter. I still get good carrots (well over 30 pounds in 2011 from two 3x10' beds, mostly good quality, Scarlet Nantes and Scarlet Keeper -- both full-sized varieties). My strategy for carrots is long-term and similar to my strategy for just about everything else (i.e. managing rotation, organic matter, and nutrients).
- My rotation plan tells me where carrots are going to be growing for the next few years.
- The year before a given bed is going to be planted to carrots I grow squash. This provides two benefits:
- Squash is hungry and does very well on copious amounts of composted horse manure. You wouldn't want to dig in too much manure right before you plant carrots, but adding it the year before lets it mellow for an extra year. All the extra organic matter helps lighten the soil for the carrots.
- A squash crop helps reduce weed pressure: before the vines creep out, you can hoe regularly to cut down the little weeds and reduce the seed bank in the soil. After the plants fill out, the leaves shade the soil and make it harder for weeds to thrive. Carrots don't compete well with weeds, so this is beneficial.
- (You could use another crop to precede the carrots; I used to get good results with peas. It probably doesn't matter that much as long as you can start loading nutrients into the bed the year prior to planting; carrots don't like fertilizer added just prior to seeding.)
- I also add wood ash to the bed. This adds phosphorus, which carrots need, and it raises the pH. Be careful not to add too much, though -- and don't use lime and ash unless you know you need it. Ash has very high pH and if your soil isn't already acidic then it will become too alkali for the carrots. (Ash also adds a lot of calcium, which carrots don't specifically require a lot of, but it is helpful to other crops and you'll be adding it if you would normally lime anyway.)
- In the fall after I harvest the squash, I apply a layer of one- or two-year-old leaf mold. You could dig it in at that time or leave it on top and dig it in in the spring before planting. This is a low-fertility way to add organic matter to the soil -- and lighten a heavy clay. Worms love leaves, too. (Incorporating the fall is probably better, but I never seem to make it in time.)
- When I dig in the leaves, I sink my garden fork as deep as I can into the bed, pulling out all the rocks and sticks that I can find. (When I started this garden, I had so many rocks and pockets of gravel that I sent the soil in several of the beds through a screen to get a finer soil; that was a lot of work but it seems to be paying off now.) Other crops aren't as sensitive to soil obstructions, so you don't have to treat every bed this way, but carrots really do well when the rocks are gone. So we can't even dig fence posts on this property, but the carrots have a bed where they can go a foot deep or more before they hit rocks. (Alternatively, you could build raised beds and fill them with material.)
- I still get the occasional forked/twisted carrot. If I lift a carrot to find that it is forked, I can usually dig around that spot and find a rock that I missed... the more careful you are at removing rocks the better the results will be. Also, while forked carrots aren't something you can sell -- or feed to picky kids -- they still clean up ok and are tasty in soup or sliced onto salad.
The end result is a bumper crop of long, mostly-straight, delicious carrots -- and incredible soil. When you pull the carrots, you're left with a fine soil rich in nutrients and loaded with organic matter. Follow the carrot crop with another dose of manure, a cover crop of winter rye if you're not overwintering your carrots, and you're ready for sweet corn the next season.
It sounds like your soil is the main problem. You should avoid stoney soil or clay soil. Clay could probably fixed by digging in other material (but not manure which is too rich for root vegetables). You may want to consider using raised beds or containers as an easier soil fix.
For poor soil conditions, the shorter more rounded varieties are indeed recommended. For example there's some general advice here at the BBC:
Carrot varieties are described as early or maincrop varieties, but also either short-root or long-root varieties. These names give you an idea of when they will crop and the type of soil they're suitable for. Carrots and parsnips grow best in light, sandy soil so if your soil is heavy clay, stony, chalky or doesn't drain particularly well, concentrate on the maincrop, short-root types which cope better with these conditions.
The article also provides some UK-specific advice (minimizing carrot fly) plus sowing instructions and recommendations for pots if the soil is poor.