Is there any way to parametrize a type by a value, rather than another type?

I believe the following web page answers your question: http://pobox.com/~oleg/ftp/Haskell/number-parameterized-types.html It uses parameterization by decimal numbers -- which seem more natural to read.

It would be interesting to know if/how you could, e.g., define a concatenation function with a type like

Yes, you can -- and the page referenced above does that.