Lumineers or Implants Best with Periodontal Disease?

It could be considered "curing periodontal disease" by removing the teeth and placing implants. If implants are a plan, now or in the future, then adequate bone is needed. It is possible that the bone level NOW is ok, but won't be there later.
Depending on the situation, veneers on "long teeth" (receded gums and bone) will appear long and unesthetic ("not pretty"). Based on the overall goal, veneers may not be an option and may not last as long due to the periodontal disease.
Without seeing xrays or photos, it is hard to make a suggestion. I would speak with experienced dentists that have treated cases like yours to see what they suggest.

The answer is: it depends at the severity of your disease. If you have advanced or aggressive disease, most likely veneers will not be the best idea as you will be throwing out good money after bad. If your disease is early to moderate and you have been maintained well for a while and are committed to be treated with maintenance for the rest of your life, then you can consider veneers. Implants are an "end stage" option and should only be considered when the discussion turns to "It's now time to extract your teeth before you lose more bone". Ask your dentist or periodontist what stage you are at now and how much bone you have lost vs how much bone you have left. Waiting too long with teeth with periodontal disease can lead to more bone loss and PRECLUDE you from dental implants in the future. Don't burn any bridges with implant and don't sit around too long on periodontal disease  because once you lose a lot of bone, you will need expensive, painful and time consuming bone grafting prior to even considering having dental implants. Food of thought. 

Most importantly, you must be sure that you eradicate all active periodontal disease.  Once all the infection is gone, and disease is arrested (where it doesn't get worse) you will need to have your Board Certified Periodontist evaluate how stable your periodontal status is.  Lumineers can be bulky and may be more difficult for you to perform  proper oral hygiene, which is the opposite of the goals of periodontal disease treatment. 

Implants are definitely a great option.  However, patients with periodontal disease have higher failure rates and may require bone augmentation prior to implants placement which may increase the length and cost of treatment.  Longevity of both Lumineers and dental implants is directly proportional to patient's compliance and oral hygiene, as well as good overall health.  Bottom line, make sure your disease activity is arrested prior to any option you choose.