"These animals come from back gardens, cages, they come from awful scenarios," explained Ms McKenna.
A spokesperson for the Department for Environment, Food and Rural Affairs (Defra) said the current legislation was under regular review and those wishing to keep non-native animals would be carefully vetted and need a licence.
Wildside Exotic Rescue currently cares for almost 200 animals including mountain lions, zebra, racoons and monkeys.
Some were donated by owners who could no longer look after them while others were seized by police and local authorities.
It is currently legal to keep wild animals as pets, even those considered dangerous, under the Dangerous Wild Animals Act 1976.
The legislation requires owners of such dangerous animals to get a licence from their local authority.
Licences are granted based several criteria including safety and where the animal will be kept.
But Ms McKenna told the BBC: "Very few [wild animals] need a licence and it is those that are deemed dangerous that do need one and the rest do not.
"So out of the 38 species here, only six require me to have a licence and be inspected.
"The rest, I could do what I like with them - I could abuse them, be cruel to them or be great with them."
The current legislation largely focuses on animals deemed dangerous.
This includes many primates, carnivores, larger or venomous reptiles, dangerous spiders and scorpions.
But Ms McKenna argued this was wrong and every non-native wild animal - regardless if it was dangerous or not - should require a licence with regular inspections.
The animal campaigner also claimed the act was "mandating cruelty".
"We want to put an end to the wild pet trade," she explained.
"At the moment it is possible [to own a wild animal] because it is legal, there are very few animals a UK home owner cannot get their hands on.
"If you fancy an animal, you just bring it home and pop it in your back garden.
"It is a stupid and as cruel as that."
A Defra spokesperson said steps have been taken to increase the maximum prison sentence for animal cruelty to five years and legislation had been brought forward to prohibit primates being kept as domestic pets.