I don't think so, it's not a state park. But it's pretty cheap, $10 for the day, or $20 for an annual pass.
If you just want to go to the inlet it's $20 and that's a yearly pass just for the inlet which includes the dog park. You do not have to be a resident of Florida. A day pass to this park only is $10.
$20 annual is only for county residents, non residents are $100 iirc. $10 day pass gives you ability to reenter once that day and good at any of the beach ramps along Daytona Beach and not just the Inlet but the Inlet is the only place where dogs are allowed and can go swim and play in the water
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