Augment a Turning machine with multiple read/write heads. In keeping with a theme of never being finite when one can be infinite, let there be an infinite number of heads organized into a linear "tape" of heads.
Each head may be controlled independently, but only one head may be active at a time. The active head is selected by moving around the tape of heads with a meta-head.
Why merely have one tape of heads when you can have many, perhaps organized in a tree structure?
Consider making the head index be accessible by address, so we can rapidly switch between heads distant from each other.
While only one head at a time may be active, a compiler or simulator can notice that different heads do not conflict with each other and run them in parallel for performance improvement.
The multi-head multi-tape multi-level Turing machine with addresses.