None anytime soon, I know there is 1 in the spring and 1 in the summer for Tacoma residents.
If you know someone with a yearly pass they can get in a few people for free.
I'm not positive, but after looking at their website I didn't see any "free days". I recommend checking out Groupon for discounts
Go to a Tacoma public library and check out the family pass. They have passes for the local museums as well.
There is one free day a year and you probably don't want to go that day. The zoo announces it ahead of time. It is usually height of the summer like July 31. Kids get in free several times a year like the week before Halloween. The rest of the park is always free. (Ex employee)
No days are free
Thanks! Your answer is awaiting moderation.