Series RLC bandwidth independence from C
There is no 'reason' per se, it's just the way the mathematics works out.

The tutorial paper you mentioned in your comments (add link to your OP please) is pretty clear and thorough, it leads you by the nose from first principles to equation 1.19, which is $$B=\frac{L}{R}=\frac{\omega_0}{Q}$$

The useful part of that equation is the \$\omega\$ and \$Q\$ bit (which you omitted from your question). Both of those have L and C in them to half powers. It just happens that when when you expand them and combine, for the series circuit, you get a whole L and no C. You might as well ask why sqrt(2) times sqrt(2) is 2.

For the parallel circuit, they combine differently to give you a whole C and no L. 

Most of the time, we're concerned with fractional bandwidth, which is why Q was invented. Dealing with absolute bandwidth while varying the C of a series circuit is not something that people ever tend to do, or would want to understand quantitively and intuitively. But if you did, then that's the formula that you would use.