score:4
According to Google (sidebar):
Hours:
Saturday Closed
Sunday Closed
Monday 8am–4pm
Tuesday 8am–4pm
Wednesday 8am–4pm
Thursday 8am–4pm
Friday 8am–4pm
But you might want to email: [email protected] as at least for The Little Walnut-Gomez Peak picnic grounds hours vary according to weather conditions in winter and also daylight at times: generally open from 8:00 a.m. until sunset during spring, summer and fall.
On the other hand, Lonely Planet shows The Purple Onion's opening hours as: 9am-5pm Fri-Sun May-Oct
so the above 'office hours' cannot apply to the park as a whole:
Mogollon, a semi-ghost town, lies 4 miles north of Glenwood and then 9 miles east on steep and narrow Hwy 159 (inaccessible during the winter). Once an important mining town, it’s now rather deserted and empty, inhabited by only a few antique-like shops and, as is typical for middle-of-nowhere New Mexico, one proud little restaurant. This one is called the Purple Onion and it claims to serve the best burger in New Mexico. We’re not sure if it’s the best, but it’s pretty darn good, as are most of the homemade dishes at this friendly, weekend-only place.
Hence by deduction from the absence of any other 'opening times', apart from weather, it is open all hours. The Purple Onion's customers must access it somehow.
Note: If you go into Mogollon, it would be wise to try to come or go at at a time when the weather/roads are dry and it is broad daylight. The road back down the mountain is one lane at times, sharing the same lane between both directions of traffic, and it will be like that around sharp 180 degree curves around the mountain with almost no visibility ahead.