A tranquil, shaded oasis awaits on the border of Tarrytown, where peacocks perch on rooftops and sashay through ornate gardens. The birds are one of the many reasons that Mayfield Park, a nature preserve near Lake Austin, feels enchanting and serene. It’s common to see brides-to-be poised near water lilies and toddlers scuttling after birds (all being captured by cameras, of course); many locals have visited Mayfield, but few know …