Depending which gate you want to use in San Ysidro. For the East gate, there is parking next to Jack in the Box and across the street from it. For the West gate, there is a large parking across the street on Camino de la Plaza.
Only you cross whit boarding passes directly to airport
I'm not sure if I understand your question. You posted on the Cross Border Xpress, which is located in Otay, not San Ysidro. There is plenty of parking at the CBX. There is also plenty of different lots to choose from at the San Ysidro border crossing.
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