I found this thread helpful when I first started playing Guns of Icarus, so I figured I'd contribute the tricks I've learned since then that aren't already mentioned.
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GalleonBalloon to Main Engine: Looking forward and standing slightly right of the balloon, jump backward over the rail, then move forward while falling. You'll land on a metal platform with the main engine on your right side. This is useful for forming an efficient circuit of the balloon, main engine, and hull.
Main Engine to Turning Engines: From the position you landed in the previous jump, walk forward until you hit a corner formed by a black support meeting the main engine. Strafe left slightly and fall off the edge. You're now on top of the port turning engine and can easily hop off onto the bottom deck. Don't use this jump just to get from the hull to the turning engines; it doesn't save time over the stairs.
Middle Deck to Aft Medium Guns: Hop off either side of the ship onto the top of an aft medium gun. Looking towards the ship, walk backward off the gun until you're falling. Wait until the screen just starts to dim, then mount the gun. If performed correctly the screen will appear dimmed while using the gun, and upon dismounting you will reappear by the hull instantly. You can sort of do this with the forward medium guns, but upon dismounting you have to wait for the full duration of the fall to reappear. You can also do this with the port light gun, but it's relatively useless. For convenience’s sake, I will use the phrase “fall mount” to refer to this technique in the future.
GoldfishBalloon to Hull: Rather than going back down the stairs, walk forward from the balloon through the overhang and fall off the starboard edge behind the helm. Fairly obvious, but I didn't see it mentioned, and it saves a good second or two.
Fall Mount Light Guns: Choose a light gun. Facing it, slowly walk away from the ship's center. You shouldn't have to fall for your screen to dim. Once it is dim, mount the gun. This is useless due to the structure of the goldfish; you will reappear in inconvenient positions.
Medium Gun to Helm: You're on the medium gun, your captain just disconnected, you're about to crash into a mountain, and you're the last remaining person on your team. Well shit: you're having a bad game. Anyway, facing aft, jump onto the black fence on either side of you. Hop across the two armor plates and then onto the black support beam. From there you can mount the helm and prevent your crash into the oncoming mountain. Had you used the stairs, your ship would be a pile of rubble.
JunkerDifferent Hull Angle: Standing on the ramp that leads to the bottom deck, you can repair the hull from below. This isn't parkour, but since it is useful information for creating circuits I'm including it. I won't include the more commonly known angles, but I will include the ones that weren't intuitive to me at first.
Fall Mount Middle Deck Light Guns: Hop on top of either of the light guns. Face the center of the ship and walk backward off the gun. Mount the gun when the screen is dimmed. Once again, this is relatively useless. You can sort of fall mount the light guns on the bottom deck and the front light gun, but you have to wait the full fall duration to reappear.
PyramidionHull to Main Engine: Facing aft, use the rail that's forward of the hull to jump onto it. Leap from the hull onto the main engine platform. Useless since you can hit the main engine from the bottom anyways, but I had to put something under pyramidion.
SpireFall Mount Medium Gun: Using the technique described in Hamster's post, get on top of the medium gun. Facing aft, walk backward until you fall off the gun. Mount the gun when your screen starts to dim. Arguably useless since you reappear at the helm (right in front of the pilot's face too
.) You can fall mount the port and starboard light guns using the technique described under junker. The fall mount for the port gun is potentially useful.
Top Deck Starboard Light Gun to Hull: Before you mount the gun, get on the aft side of it and stand on the silver metal plating that is adjacent to the black fence. When you dismount, facing aft walk forward and fall onto the spiraling black fence below you, then walk forward again. You should fall just next to the starboard turning engine. You can do it differently, I just find this is the most reliable way of not falling off the ship.
SquidDifferent Hull Angle: You're the pilot, fleeing an enemy, when all of a sudden your hull armor disappears! You look over to see where your engineer is, but he disconnected and the god damned AI is repairing the port turning engine instead of fixing the armor. Deep breath. Dismount the helm, turn starboard and jump over the rail onto the slope. That's it! You can repair the hull through the bars here then get back to flying.
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That's all I've got. As for my take on “Fall Mounting: Bug or Feature?”, I think it provides a minor buff to ships that don't see enough use, so there's no harm in letting it stay. I hope a few people learn something new from reading this, and contribute back to the thread when they find a new trick.