Is there a specific way to describe over-grown, old, tough vegetables?

picked past their prime (also) peak

• When harvested past their prime, beets have a strong taste and a tough, pithy texture"
  (Google Books)

• Here are some ways that you can use your frozen, past-prime fruits and vegetables.

• Tomatoes that are past their prime can be pureed and frozen for up to 6 months.
  (Frugal Living Mom.com)

• The Pilgrims may have eaten turnips, but why should we? Let's face it; the turnip can be a tricky vegetable. If they're even a day or two past their prime, you'll wind up dining on a veggie that tastes more like a piece of wood than an appetizing holiday side.
  (How Stuff Works.com)

• When florets [broccoli] on the outside edge of the head are large and full, this is another helpful indicator that you’re good to go. But don’t wait too long. When they begin to turn from green to yellow, this is a sign that they’re beginning to flower, and past their peak.
  (Gardener's Path.com)

A single-word alternative could be

overmature

overmature: past the age or condition of maturity
a : beyond the stage of desirable or optimal development or productivity

• overmature wax beans

• The squash is very tender and should be handled with care to prevent skin cuts and bruising. Avoid purchasing any with … a dull, shriveled skin, which is a sign of overmaturity.
  (Merriam-Webster)

The scientific term for this process is senescence.

This word encompasses the aging of plants in general. It describes not only the phase that fruits and vegetables enter after their ripening phases, but also the process by which leaves turn colors in the autumn, etc.

This is the wikipedia article on the process of senescence.

The adjective form of the word is senescent.

Showing signs of the farmer's neglect, the fields were filled with overgrown vines and vegetables in various states of senescence.