Having seen the male rats in the local pet stores, I’m suitably impressed. I can’t imagine anything having testicles much larger relative to body size – they wouldn’t be able to move.
I assume you mean largest relative to body in a normal condition. Some people with elephantiasis have outrageously enlarged testicles that certainly does impede motion.
There is a fish species, the male of which attaches itself to a female’s body, links into her blood supply, and atrophies down to being merely a circulatory system and testicles. Consequently the testicles are at that stage the majority of its body-weight. I think that’s pretty much unbeatable, except I have no idea of the name of this species sorry.
Anglerfishes perhaps? They have an incredible amount of sexual dimorphism and males act as parasites much like you say. Wikipedia is sorely lacking in recording 'nad size, but the male apparently has large testes (BBC).
There’s some kind of marsupial rat-like creature in Australia which only meets female of the species very occasional. Thus they store up impressive amounts of semen in the testicles for those special but rare occasions.
The Honey Possum Tarsipes rostratus, a tiny marsupial that feeds on nectar, is listed by at least one site as the mammal with the largest testes in proportion to body size, but since they “only” amount to 4.2% of body weight, that is exceeded by the bat listed above.
Boars have big testicles. I don’t know what percentage of their body weight they amount to, but head over to the Swine barn at your local county fair and you’ll see 400lb boars with testicles the size of basketballs.