What items could you bring on-board to maximise your chance of survival in an emergency?
Most of the items you can bring with you to maximize your chances of surviving an in-flight emergency are items you're not going to be allowed to bring with you.
About the only things I could recommend bringing along on a commercial flight are your cellphone and a personal locator beacon (PLB) of some kind, but frankly if your airliner goes down ATC is going to know where you were when they lost contact, (and the plane has an Emergency Locator Transmitter already).

You should also wear sensible clothing -- natural fibers like wool or cotton are the order of the day if in-flight fire is a concern. Polyester or other synthetic fabrics will melt to your skin and cause you to have a Bad Time.



More directly addressing your question though, the items that spring immediately to my mind (and why you probably won't be taking them with you) are:  


A parachute (what prompted your question)
Sure, you can bring your parachute. It's kind of a useless security blanket though.
From cruise flight you'd basically be executing a high-altitude jump (let's say a HALO jump, because you want to get down as quick as you can). You would require oxygen to remain conscious longer than about 30-45 seconds (though an automatic-opening chute may save you here), as well as protective clothing to survive the low temperatures while you're free-falling.
Those concerns aside, remember that airliner cabins are pressurized and the doors are typically "plug doors" that open inward. No human is strong enough to open that door with the cabin pressurized, and if you've lost cabin pressure for some reason you're better off putting on that little drop-down oxygen mask like the nice flight attendant told you to in the safety briefing. You're not going to save yourself if you pass out from hypoxia while putting on your chute and trying to get to the door.
An oxygen mask / smoke hood can be useful in a fire.  


Most airlines are not going to let you carry an oxygen cylinder onto the plane (as others pointed out in comments, oxygen and in-flight fires can cause more problems than it solves).  That holds true for pretty much any kind of compressed gas because really, how do they know what's in that tank?  
A smoke hood using a filter (or a sealed rebreather hood) with no compressed air supply may make it past security, but ask yourself honestly: "Am I proficient enough to use an emergency escape smoke hood in an in-flight emergency?" -- If you can't answer "yes" you're probably putting yourself (and others) in more danger wasting time donning the hood versus just getting out of the aircraft.

A life vest (if traveling over water)
This is overkill on an airliner (though those of us that fly little planes should seriously consider flotation devices when flying over water).
Your airline seat cushion really CAN be used as a flotation device. Those big yellow slides also detach from the plane and make marvelous floats. Frankly if you need any of these items you're in a situation where having your own Mae West isn't going to really enhance your chances of survival.
A utility knife capable of cutting your seatbelt
This is incredibly useful for a bunch of reasons, not just cutting your way out of your seatbelt. In fact I keep one in my plane. Sharp and/or pointy things make airport security nervous though - if you try to take such an item on an airline flight it will likely be taken away from you before you get anywhere near the plane.
Food and Water
Useful if you have to make an emergency landing in a remote area, but frankly airliners already have beverages (some may even still have food).
If you're carrying a light snack you're as set as you're going to be in this department.
Emergency Blankets
Nice to have, but most commercial aircraft already have shock blankets on board (and much like food/water, this is really only useful if you have an otherwise uneventful landing in a remote area).
If it makes you feel better to have your own blanket you can get pre-packed ones that can fit in basically any carry-on for about $5-10.
Signal flares
Again, mainly useful if you have an emergency landing in a remote area (and your PLB and the plane's ELT don't work for some reason). Flares are good both for signaling rescuers and for starting fires to keep warm.
I don't know if rescue flares are still routinely carried on commercial airliners, but I can say with reasonable certainty that you will not be allowed to carry on your own. They don't let you smoke on most airlines these days - they're certainly not going to let you carry on a self-igniting flare!