There is no disease, no other problem. You can still eat fruit like this, though you might decide you don't like to eat the less-developed parts.

These types of cucumbers are very susceptible to curving. Thrips are a common cause. Others include slight scarring from
tendrils, leaves or other mechanical damage.

At least for me, zucchini is kind of a benchmark in terms of growth - one plant grows huge. Pole beans are similar - lots of production. Then cucumbers, which are a bit slower and more difficult. Bell peppers are much slower and smaller.

While some parthenocarpic cucumbers are burpless, the presence (or lack of) seeds is unrelated to whether or not the cucumber is burbless. Many burbless cucumbers are long Asian varieties which develop seeds.

If the cucumber in the photo is of a parthenocarpic variety, then it should develop normally regardless of pollination, and there are other issues causing the curling (which is my suspicion anyway). If it is a conventional cucumber requiring pollination, then it is possible that lack of pollination could cause deformity. In my experience, though, cucumbers tend to abort unpollinated blossoms very quickly, at a much smaller size.

My plants develop cucumbers that curl like that when they are under stress (due to heat or lack of water), or late in the season when the vines have already been producing for a long period. It appears that there is mulch around the plants, which would be my first recommendation. They might respond to more water, and to a light application of fertilizer... but if the curling is being caused by heat, then new cukes should begin to develop normally with a return to more moderate temperatures. Until then, pinch off all deformed fruits.