Set a parent shell's variable from a subshell

The whole point of a subshell is that it doesn't affect the calling session. In bash a subshell is a child process, other shells differ but even then a variable setting in a subshell does not affect the caller. By definition.

Do you need a subshell? If you just need a group then use braces:

a=3
{ a=4;}
echo $a

gives 4 (be careful of the spaces in that one). Alternatively, write the variable value to stdout and capture it in the caller:

a=3
a=$(a=4;echo $a)
echo $a

avoid using back-ticks ``, they are deprecated and can be difficult to read.

There is the gdb-bash-variable hack:

gdb --batch-silent -ex "attach $$" -ex 'set bind_variable("a", "4", 0)'; 

although that always sets a variable in the global scope, not just the parent scope

You don't. The subshell doesn't have access to its parent's environment. (At least within the abstraction that Bash provides. You could potentially try to use gdb, or smash the stack, or whatnot, to gain such access clandestinely. I wouldn't recommend that, though.)

One alternative is for the subshell to write assignment statements to a temporary file for its parent to read:

a=3
(echo 'a=4' > tmp)
. tmp
rm tmp
echo "$a"