The seller of this June 18, 1962, photo claims this was South Padre Island, but the original script on the back of the photo labels it Padre island, which is north of South Padre (I know, that location may seem obvious). According to Wikipedia, South Padre was closed until 1962, which makes it more likely that this is Padre and not South Padre. That said, I’m no expert on this region.
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“1962 Press Photo Jeep “Off Roading” on South Padre Island Sand Dunes, Texas. This is an original press photo. Jeep rolls across sand dunes of Padre Island in Texas. Photo measures 10 x 8.25 inches. Photo is dated 06-18-1962.”