@Jonathan Hawkins any board member can delete a card, however if you have a board that is part of a business class team you can add people as "observers" which restricts them to being able to comment only.
I think everybody in the board can delete a card.
But with the Business plan, you get to have Advanced Admin Permissions, check link below
https://help.trello.com/article/710-setting-up-your-business-class-team
for more pricing and features, check link below
https://trello.com/pricing?&aceid=&adposition=&adgroup=105703213888&campaign=9843285526&creative=432679446174&device=c&keyword=trello&matchtype=e&network=g&placement=&ds_kids=p53016482445&ds_e=GOOGLE&ds_eid=700000001557344&ds_e1=GOOGLE&gclid=CjwKCAjw8J32BRBCEiwApQEKgdL_8cwmxJ4FzGuSdJMo79QB59LH1tiWJ5IQIKEJPWrVN8q96wje4hoCwj8QAvD_BwE&gclsrc=aw.ds
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