I'm Dr. Karing and I would be glad to help with your question about your cat Parker. First, it is important to clean the affected area of carpet with an enzymatic cleaner to destroy the compounds causing odor (i.e. which remind Parker of his deed). This one is my favorite:
Next, make sure Parker has at least two litter boxes (uncovered are preferred since elderly cats often feel vulnerable in enclosed dark spaces) that are kept very clean (scoop twice daily at a minimum). If you have more than one cat, then the number of litter boxes should equal one more than the number of cats you have. Hyperthyroid cats often produce large volumes of urine (i.e. sometimes they have diarrhea too) so keeping the box clean enough for him can be a challenge. An extra large box is ideal so do consider one of these storage containers for about $20: https://www.walmart.com/ip/Sterilite-66-Quart-Ultra-Storage-Box-Stadium-Blue-Case-of-4/44785815
Please also consider using Cat Attract Litter in at least one of the boxes. It has a herbal component that makes it very appealing to cats:
Amazon has the litter for a very reasonable price with free delivery if you use Amazon Prime:
Now that we tried to make the litter box as appealing as possible (i.e. clean, large, multiple uncovered boxes, special litter) and removed all evidence that remains in the carpet, the next step is to make behind the couch a poor choice. I would suggest placing a food/water dish behind the sofa: cats naturally don't want to soil areas where they eat and drink. You can also try other deterrents such as a carpet runner turned upside down so that the prickly side faces up (i.e. that isn't as nice as the food and water dish approach but may get the job done).
Feliway pheromone diffusers strategically placed where Parker spends the most time can help if stress is a part of the reason why he chooses to do this.
You can also try adding a night light around the litter box area since older cats often have diminished vision in the dark.
If all this fails, I would suggest having Parker's blood pressure checked since hypertension is common in hyperthyroid cats and can contribute to odd behavior and check a urinalysis to evaluate for an infection (i.e. less likely since he only exhibits the behavior at night but worth checking if the problem persists).