Paronychia is an infection of the skin on the sides of the nails. Typically there is swelling and pain and redness that is localized to the tissue right on either side of the nails. If you have symptoms like this, then a paronychia is definitely a possibility.
However, there are other possibilities as well. Especially in people who have their hands in water all the time, chronic irritation of the skin is very common, and this can even sometimes be eczema
of the hands. One major simple step you could take is to make sure that you always wear rubber gloves while washing dishes, as this will reduce the inflammation and irritation. You should also make sure to moisturize your hands regularly, especially after they get wet.
In addition, I recommend that you see your primary care doctor
, especially if the symptoms are persistent. First of all, they will be able to confirm whether or not you have paronychia, which might require medical treatment, such as with antibiotics. If, on the other hand, this is eczema caused by chronic irritation from dish water, you may need a strong steroid cream to help clear up the inflammation and itching, and your doctor will be able to prescribe this for you.