I have 8 dining room chairs. Two have wiggly backs. They are parsons chairs. When I investigated I found that one of the rear legs/back supports is not a solid piece but pieces together just higher than the seat. What kind of bracing could I use to strengthen it? It won't show under the upholstery so a metal piece perhaps?
That's known as a "finger joint" and it's in the worst possible location for the back support. It is almost impossible to separate it and use any dowel up through the center, not a job for the inexperienced either. Your best bet is to use metal mending plates screwed into the front and rear of the leg support with 1/2" to 3/4" long screws, probably a number 6.The plates should be as wide as the wood and have at least 3 screws per top and bottom on each plate for a total of 6. The plates will need to be bent to conform to the bend in the wood, no big deal using a vise and a mallet, but check for proper fit each time as you bend it slightly.
If you can separate the joints far enough to brush epoxy in between the fingers, that would be good! Do NOT use glue, only epoxy as it's the only adhesive that will adhere to old glue or other adhesives. Align the pieces as close to original as possible. You don't need clamping pressure with epoxy, just secure it from moving with what ever you have on hand..... strong tape etc.
If you understand what happens to the structural forces when you lean backwards, the front face of the joint is in "tension" the rear face is in "compression" and the wood in the very center is in equilibrium. That's why I recommended putting the plates on the front and rear faces, the front being in tension which is best for the metal plate. The screws will have to pull out for it to fail. If you put the plates on the sides, the screws would have to shear off, for it to fail. The best location structurally is certainly up for debate ......
I would go ahead and reinforce the other chair backs, if only by putting a mending plate on the front face where there is the most stress/force.