You would have to ask permission from the park to find out if you could or not...each park probably has different policies. I have seen some parks and some parking lots for stores be used as yard sales for fundraisers.
If there is an area of grass in front of the apartment you could ask the managers for permission to hold a yard sale there.
If you have any friends with houses try and see if you can hold a yard sale together with them in their yard. Or if they will let you borrow their yard even if they don't want to do a co-yard sale.
You could take your stuff to a flea market and rent a spot.
As someone else suggested, you could do an "indoor yard sale" or "tag sale" in your apartment. Some people don't like that though, mostly for safety reasons, especially if you are a woman (make sure another person does it with you if you choose this option). And things you don't intend to be sold people could pressure you into selling or even steal...they could even use the opportunity to be in your apartment to "case the joint" or a later robbery or such. I don't believe in being fearful, but simply aware of the possibilities. I have a friend who had an indoor yard sale and it worked out well for him.
Think outside the box and see if there are places you can rent indoors or outdoor spaces for use. I know a lady who rented a retail space that had remained empty for a very long time to hold an estate sale of her mother's belongings...the sale lasted about a month instead of a typical weekend. I have also seen people hold "mini" yard sales in front of the RV Parks where they are staying.