Why is the asymmetry of my eyes getting worse? (photo)

You have upper eyelid retraction.  The most common cause to this is thyroid related orbital disease.  Your internist can test your thyroid function.  The eyelids should be assessed by a fellowship trained oculoplastic surgeon.  They can also discuss with you how this is managed.  I also have information regarding this in my free ebook on eyelid surgery cited below.

See an oculoplastic specialist or ophthalmologist for eye evaluation. You may have thyroid eye disease or other, causing eyelid retraction and possibly bulging eyes. See link below.

I notice quite prominent eyes with the left side in the picture being worse. There is retraction of the upper lids and the scleral white is showing under the upper lid. This is usually not the case; in fact, the upper lid should cover a little of the iris. Sometimes, thyroid disease can be responsible for this and you want to evaluate with your physician. 

You have eyelid retraction worse on the left.  This means the lid is too high and you can see white sclera above the cornea.  While you may have had this all your life, it can also indicate possible thyroid issues - especially if you have noticed it worsening.  See an oculoplastic surgeon for an evaluation of your eyelid muscles. 

Judging from the one photo, it appears that your eyes are bulging, especially the right. Thyroid-related Graves' disease is the most likely cause. You should consult an Oculoplastic (oculofacial plastic surgeon) regarding this condition. Non-surgical option including Botox and specialized eye drops could help you. Surgery should be reserved for vision threatening situations.