Sometimes yes, and it depends on the park and how professional your setup looks. Many pretty public parks draw a line between a casual phone snap and a shoot with tripods, lighting, or a crew.
According to the
Orlando Economic Partnership, film and photo permits are free in many area jurisdictions unless city, county, or state services are involved, like off-duty officers, public works, or metered parking. To shoot on public property, you'll typically need a general liability policy of $1,000,000 that names the city or county as additional insured. Individual cities and parks set their own fees and timelines, so confirm with the specific park before you show up.
Pro tip: Book a private space through Peerspace and you're shooting on private property with an established host, which lets you skip the public-park permit process for that spot. Large productions may still need permits and insurance, and staying compliant is up to you and the host.