Is anyone able to settle an ongoing debate regarding the use of commas in lists when brackets are also involved?

When my mum proof-reads my stepdad's books and articles, she always adds commas after the brackets in the lists. The brackets usually refer to Latin names of animals. It's a bit difficult to explain but it should become clear in the example:

"The cats seen included the lynx (Lynx lynx), jaguar (Panthera onca) and leopard (Panthera pardus)."

However, my stepdad removes the comma after (Lynx lynx) on the basis that the closing bracket offers enough of a pause in the sentence to render the comma unecessary. I agree with my mum on the basis that without the comma the sentence would be clumsy and nonsensical, were the sections in parentheses to be removed.

Who is right? Or is one of these 'it's all down to writing style' matters?

Thanks.

I agree.

Even without the brackets, it's still a list, and should be punctuated as such:

The cats seen included the lynx, jaguar and leopard.