Will Sun Exposure After Chemical Peel Cause Permanent Tanning?

Chemical peels leave your skin sensitive and vulnerable to damage. During recovery, treating your face with the proper skin care products and wearing protective sunscreen will ensure you do not hamper your results and prolong your recovery. Sun exposure is responsible for much of the damage that chemical peels work to eliminate, so prolonged exposure to the sun immediately following a chemical peel, even with sunscreen, is not recommended.
Your best bet to fix the damage is to continue following your doctor’s advice. Without photos or more knowledge of your skin’s condition, making a different recommendation than what your doctor has prescribed would be difficult and ill-advised. If you have any further questions, be sure to talk them over with your doctor. Hope this helps.

Sounds like your Dermatologist is doing just the right thing with a combination of a bleaching agent (hydroquinone) and a topical corticosteroid.
You need to be cautious about sun exposure soon after a peel to avoid prolonged redness which can then lead to post-inflammatory hyperpigmentation, and if severe, (but unusual with a medium peel), permanent scarring.

Be cautious with any further sun exposure.
It typically takes somewhere between 3-4 months for this to resolve. Some areas of hyperpigmentation may take longer (up to two years).