Can urban parks bridge social divides, or do they reinforce them? This study leverages novel mobile device data to analyze "experienced income diversity", which are the actual encounters between socioeconomic groups, within New York City's parks. This study finds that parks are ambivalent spaces: while holding potential for integration, they often reproduce segregation. Spatiotemporal analysis reveals that mixing peaks on spring afternoons but is predominantly accessed by middle- and high-income groups. Lower-income populations experience more homogeneous environments. Critically, park amenities drive this stratification. Facilities like playgrounds foster cross-class interaction for the poor but self-segregation for the rich, while sports courts and water features primarily serve the wealthy. The findings provide a data-driven framework for targeting park investments to proactively foster social integration, moving beyond simple amenity provision to actively manage for equity.