Shelves and Cabinets
Start with hard-to-reach shelves or cabinets placed too far up on the wall. If you can make the items in those cabinets easier to access, you'll reduce the number of times your parent must stand on a stool or chair. If there's no way to eliminate the high storage space, at least move most-used items to the lower shelves and make sure a sturdy stepladder is on hand. Many small stepladders are designed with extra-wide treads and easy-to-grab supports, and some fold down easily to fit in small closets.
Next, take a look at where the microwave oven is placed. The people who use it should never have to reach above eye level to take out food -- a good policy for any kitchen, but an absolute necessity for those whose hands may not be as steady as they used to be.
If the microwave is built into a base cabinet below the counter, you may want to move it up to sit on the countertop. Even though a low-placed microwave doesn't pose a serious safety risk, it does require you to bend over to use it.
The dishwasher also requires a lot of bending; if possible, try installing the appliance on a 6- to 8-inch platform to reduce the reach when loading and unloading. This trick also works in the laundry room to raise the clothes dryer to a more comfortable height.
Knobs and Handles
Swap hard-to-grip small knobs for C-shaped or D-shaped cabinet door and drawer pulls. This type of pull is a plus for any kitchen and can help take the pain out of cooking for someone with arthritis.
Apply the same logic to the kitchen sink, and put in a lever handle instead of knobs that require twisting.
While you're at it, think about switching doorknobs throughout the house to lever latches -- they're easier to use and look good too.