You sound like perfect candidates for MRI's new "The Park" project, currently under construction in front of and around the Monte Carlo which, when completed (likely some time next year), will comprise an outdoor promenade/piazza intended for congregating, chilling out, and catching some low-key al fresco entertainment, right at the heart of the Las Vegas Strip.
The fact that this project is currently under construction is indicative of the fact that there isn't really anything similar to what you're looking for that currently exists: While Las Vegas is actually home to a lot more green spaces than you might think, these are pretty much all located away from the casino hubs in the residential hinterland, so you're going to need a car to get to what you're looking for.
Of the park choices Las Vegas has to offer, Sunset Park is the largest park and our top pick, located just east of McCarran International Airport, at Sunset and Eastern. Described as "the crown jewel in the Clark County's park system," it's been serving the entire Las Vegas Valley since 1967, with phased expansions over time having developed 185 of the 323 total acres. This amenity boasts a decent-sized lake, which is used to full potential during the annual Renaissance Fair that it hosts, plus a recently expanded dog park, tennis courts, softball fields, walking trails, a children's play area, numerous species of rare plants (and wildllife), and the free parking and restrooms that you inquired about.
As far as the downtown area is concerned, the nearest option would be Freedom Park, located on E. Washington, between Mojave and Pecos. This 68-acre facility boasts paved bike trails and walkways, real grass with plenty of big shade trees for blanket picnicking, a turf football field, basketbaall courts, soccer pitch, skate park, baseball park, bocce ball and horseshoe courts, playgrounds, picnic tables with shaders, a concession stand, and restrooms. It's clean and safe by day, but we wouldn't necessarily recommend venturing there after dark.
As you can surmise from all the facilities there, Freedom Park is more of a community-activity facility than a chill-out zone; for something more akin to what we think you're looking for, we recommend traveling further north to Floyd Lamb Park, a well-maintained locals secret, with its multiple fishing ponds, plenty of grass and shade trees, free-range fauna (including ducks and peacocks), and dog-friendly policy. Check out this photo gallery for a good overview of what to expect. Parking is plentiful and free, and restrooms are available.
One final suggestion, which is even further off the beaten path but well worth a visit, is the Las Vegas Wetlands Park, which is another "work-in-progress" and has recently undergone some significant expansions and upgrades. It currently encompasses 2,900 acres, filled with trails and wildlife, and is intended specifically as a counterbalance to the loss of wetlands elsewhere. It's official address is 7050 Wetlands Park Lane, but basically just head east on Tropicana until you run out of road, and you've arrived. The park is open daily (major holidays excepted) from dawn to dusk, while the Visitor Center closes at 3 p.m. Admission is free and there's paved parking, restrooms, and shaded picnic tables.
