Caught another Shiny Magikarp...
I didn't count exactly how long the chain was, but I know that it was greater than 20, but most likely not greater than 50, which supports the theory of shorter chains being more fruitful.
Now, at last, I shall experiment with the Good and Super Rods at Parfum Palace.
EDIT:
Shiny Goldeen at Parfum Palace, standing in the exact same spot (not that it matters; just want to keep all the factors the same except for the variable), using a Good Rod. With the Good Rod, my chain broke twice because of "no bite" before I found a shiny; the chain was 23.
After I do some testing on the Super Rod, I'll keep going with the Old Rod in the same spot and see if I get any chain breaks because of "no bite."