Call Balcones Recycling at (512) 472-6200 to arrange a pallet pickup from a business. They will pick up metal, wooden, and plastic pallets during business hours for no charge.
If you're wanting a home pickup, the company may be able to do that if you live inside the city limits. If you're outside the city limits, they will not do home pickups.
Other options for only wood pallets are Roadrunner Pallet Recycling at (512) 990-1090, and the Austin Pallet Company at (512) 990-0090, both of which will pay for wood pallets in good condition.
Take the pallets to the recycling center if they can't do a pickup. Take the pallets during business hours to ensure they're handled correctly.
Contact the city of Austin if you're unable to transport the pallets. In an effort to keep the city clean, they will sometimes help you make arrangements to have large items moved from your home.