H, the mountain and immediate area is accessible by anyone, but a pre booked guided tour is required to get beyond the gate and onto the top of Paranal itself. The tour is free and can be booked here:
http://www.eso.org/public/about-eso/...s/paranal.html
How to get there:
http://www.eso.org/public/about-eso/travel/paranal.html
The nearest city is Antofagasta on the coast 120 km to the north. The drive took a couple of hours. The road to the observatory is apparently fully paved now, but when I was there in 2008 they were still working on it.
I've attached a few photos from our trip to illustrate the experience. If you are ever in the area it is an absolute must see.