Just over the line
Everything else here is Delaware. These aren't. The state is forty miles wide, so for a great many people in New Castle County it is closer to somewhere in Maryland or Pennsylvania than to Sussex — and the distance is still the first thing you need to know.
From these pages
Nonprofit zoo · Rising Sun
About 25 minutes west of Newark on Telegraph Road, and over the Maryland line.
A small nonprofit zoo in Cecil County, close enough to Newark to be an easy morning. Giraffes, tigers, bears, wolves, zebras, macaws and tortoises, plus mini golf. Peacocks and chickens wander the paths rather than sitting behind wire.
Seasonal — see dates→
Working water-powered mill and 160-acre preserve · Glen Mills
About half an hour north of Wilmington up Route 202 and over the Pennsylvania line, near Glen Mills.
A mill built in 1704 that still grinds with water, sitting in 160 acres of preserve with a stream, ponds and a mill race running through it. The grounds and the trails are free — the mill tour is the part you pay for, and the two are priced separately. Wooded creek walking rather than open park: shade, stone crossings and a lot of water.
Season not confirmed→
This site says it covers Delaware, and that's worth something precisely because it's narrow. Filing a Maryland zoo in with the state parks would spend it. You don't care about state lines. A reference that tells you what it covers has to.
They're a different kind of record too. A private business sets its own prices and changes them without telling anybody, where a state park's fee gets published by DNREC. So expect admission to be the first thing that goes stale here. Call ahead.