In the social menu, you have option of visiting a random player, similar to the "visit" option for friends on your list (assuming you have any). Other than shame and self loathing and the inadequacies of my fledgling dragon park, can I get any benefit from visiting a random park?
There is no gameplay benefit. You can not get coin bonuses, gems or anything else from the random player, nor can you see their name or invite them to be friends in any way.
Visiting random players can be a great way to see the depth of the game and beautiful parks created by other users. You can see all sorts of items you haven't yet unlocked which can serve as hints to what you can later accomplish, or you can get neat ideas of what to do with your park.