Parallel leg riding was the bread and butter of Seth Ciasulli's (Red Hawk's owner) top riding game. Seth was extremely good at it and has pinned plenty of wrestlers in the top ten in college. We at Red Hawk are part of the very few who teach this series and teach it correctly.
The real goal is the power half which is a very powerful move and is capable of pinning the best wrestlers. The biggest enemy with this move is the referee. You must maintain good pressure while keeping the referee happy. It's a tricky line to walk, but if you can, no one will stop you.
There are priorities you must stick to in order to be a good leg rider. If any of these come undone you must start over in your priority list and then move forward again. If you go out of order, you'll likely get reversed. Keep your hips on top at all times. Keep your feet locked up. Break your opponent down. Maintain good hip pressure. Work to turn. If any of those come undone you have to correct it before trying any moves.
Please, don't learn this the hard way. You'll get out of position and lose matches against better wrestlers. It always helps to listen to the advice of someone who has been there before you. That's called being coachable, and it's a very important quality to have if you want to be successful in life.
You'll have two legs in. You will be sitting on top of his hips or lower back, with your feet wrapping around the side of his hips and through his legs. You should have both feet locked up at all times to make it more difficult for your opponent to just slide his legs out. This will also allow you to put more hip pressure. You should work to flatten your opponent. If he's on his base, you can do this by grabbing one wrist with two hands and lift it forward.
If he's on his stomach yet propped up on his elbows, you can reach on the inside of his arms and grab the wrists, and pull them backwards. This is referred to as rolling the wrists out. Hip pressure means your hips will be down, you'll be leaning forward, yet your hands, head, and chest are all elevated. This puts all of the weight north of your hips placed directly where your hips are, which makes you very heavy. It will also lift up his legs and suspend them in the air. This makes it very difficult for your opponent to come up to his base.
You will probably have to break him down constantly as you look for moves. Be sure to stay busy. If you aren't doing much, or are only trying one move over and over, the referee will call a stalemate and start you over. If this happens enough you'll get called for stalling. It's not really on the bottom guy to improve, it's on the top guy, because if you're doing your job right in this position, your opponent can't do much to get up.
If you grab his wrist and pull it back and he doesn't roll it out, he stuffs it under his chest, you can grab it with both hands and lift it into the air. You can use your right arm to Under-Hook the arm while putting your elbow in his back to get more leverage. This will let you thread your left arm under his left arm and across his back, giving you an Arm Bar.
Instead of getting an Arm Bar, you can take the wrist and place it across your opponent's back. This is called a Hammerlock, or we used to call it a Cassidy Ride. Make sure the hand stays at the belt line and on his back. If the hand starts to wander up towards the shoulders the referee will call potentially dangerous and reset you. If you pull the wrist away from the back it's actually called illegal. The referee will give your opponent a point and reset you. Once you place the hand across his back you can apply hip pressure to secure it and then you'll have two hands left to manipulate the right side of his upper body.
Defensively the best thing you can do is keep legs out or take the legs out when they're going in. If they're already secured, then it'll be tough to do anything. One of the best things you can do when wrestling an experienced leg rider is to keep him in a Crab Ride. The Crab Ride itself is part of a leg rider's arsenal, but it is the most unstable part and a lot of leg riders are not very proficient at keeping your hips controlled. They're also usually not the best at judging when to bail when in a Crab Ride.