Q:

How to redirect an output file descriptor of a subshell to an input file descriptor in the parent shell?

(In BASH) I want a subshell to use a non-STDOUT non-STDERR file descriptor to pass some data back to the parent shell. How can I do that? Eventually I would love to save the data into some variable of the parent shell.
(
  # The following two lines show the behavior of the subshell.
  # We cannot change them.
  echo "This should go to STDOUT"
  echo "This is the data I want to pass to the parent shell" >&3
)
#...
data_from_subshell=... # Somehow assign the value of &3 of the
                       # subshell to this variable

EDIT:
The subshell runs a black-box program that writes to STDOUT and &3.

A:

BEWARE, BASHISM AHEAD (there are posix shells that are significantly faster than bash, e.g. ash or dash, that don't have process substitution).
You can do a handle dance to move original standard output to a new descriptor to make standard output available for piping (from the top of my head):
exec 3>&1 # creates 3 as alias for 1
run_in_subshell() { # just shortcut for the two cases below
    echo "This goes to STDOUT" >&3
    echo "And this goes to THE OTHER FUNCTION"
}

Now you should be able to write:
while read line; do
    process $line
done < <(run_in_subshell)

but the <() construct is a bashism. You can replace it with pipeline
run_in_subshell | while read line; do
    process $line
done

except than the second command also runs in subshell, because all commands in pipeline do.

A:

The easiest way of course, is to capture the output directly in the parent 
data_from_subshell=$(echo "This is the data I want to pass to the parent shell")

You can use a named pipe as an alternative way to read data from a child 
mkfifo /tmp/fifo

now you can redirect the child to /tmp/fifo 
(
    echo "This should go to STDOUT"
    echo "This is the data I want to pass to the parent shell" >/tmp/fifo
) &

and the parent can read from there 
read data_from_subshell </tmp/fifo

Another way is to use coproc to start a child process. This creates a child with a bidirectional pipe and redirects the child's stdin and stdout to the pipe descriptors. To use both the pipe and stdout in the child, you must duplicate stdout in the parent first 
exec 4>&1 # duplicate stdout for usage in client

coproc SUBSHELL (
    exec 3>&1 1>&4- # redirect fd 3 to pipe, redirect fd 1 to stdout
    (
    echo "This should go to STDOUT"
    echo "This is the data I want to pass to the parent shell" >&3
    )
)

exec 4>&- # close fd 4 in parent
read data <&${SUBSHELL[0]}
echo "Parent: $data"

Coprocesses were introduced in Bash 4.0.

