Customize airline tagbar extension to include current tags parent

The airline plugin has an extension which uses Tagbar to show the current function name.

^{(I know this because of this question which I found via asking this question)}

This works really well and looks great. Here's airline showing me I'm editing the do_foo() function:

But Tagbar knows much more than this! Specifically, it knows the current class (i.e. the parent of the current tag):

^{(It does this using ctags)}

Is there a way to configure airline to tell me the current class as well as the current function?

Full start-to-finish process:

Install universal-ctags: apt-get install universal-ctags or alternative Add these lines in your init.vim / .vimrc, doing an update with your package manager.

Plug 'vim-airline/vim-airline'
Plug 'preservim/tagbar'
...
let g:airline#extensions#tagbar#enabled = 1                                                                                                                                                                   
let g:airline#extensions#tagbar#flags = 'f'