Henry Hudson Parkway as it cuts through Van Cortlandt Park in the Bronx. Image: Google Street View. The Henry Hudson Parkway was part of Robert Moses' West Side Improvement project, with the majority of the funding and labor from the New Deal, as explained in the references below. Moses chose to run the new parkway through Van Cortlandt Park because it was already city property. To run it outside of the park would have required condemning and buying up vast amounts of privately owned land. This image shows where the parkway enters the park at about 254th Street; the Parade Ground is seen through the trees at right.

References:

1. da Cruz, Frank, Riverside Park New Deal Sites.
2. da Cruz, Frank, The Henry Hudson Parkway.
3. Caro, Robert A., The Power Broker - Robert Moses and the Fall of New York, Vintage Books (1974), pp.526-540.
4. NYC Parks Dept press release of October 3, 1937: “The Department of Parks announces that on Tuesday, October 12th, the 6.7 miles of parkway now under construction in Riverside and Fort Washington Parks in connection with the West Side Improvement will be officially opened to the public, thereby adding another vital link in a great arterial express highway and parkway system for the City of New York.”
5. HISTORIC AMERICAN ENGINEERING RECORD HENRY HUDSON PARKWAY HAER No. NY-334, New York Metropolitan Transportation Council (2006).
6. New Deal Assistance in NYC Parks Department Projects, 1934-43.