Modern bash
If you are running bash v4 or better, the PID of the subshell is available in $BASHPID. For example:
$ echo $$ $BASHPID ; ( echo $$ $BASHPID )
32326 32326
32326 1519
In the main shell, $BASHPID is the same as $$. In the subshell, it is updated to the subshell’s PID.
Old bash (Version 3.x or Earlier)
Pre version 4, you need a workaround:
$ echo $$; ( : ; bash -c 'echo $PPID' )
11364
30279
(Hat tip: kubanczyk)
Why the colon?
Notice that, without the colon, the work-around does not work:
$ echo $$; ( bash -c 'echo $PPID' )
11364
11364
It appears that, in the above, a subshell is never created and hence the second statement returns the main shell’s PID. By contrast, if we put two statements inside the parens, the subshell is created and the output is as we expect. This is true even if the other statement is a mere colon, :. In shell, the : is a no-operation: it does nothing. It does, in our case however, force the creation of the subshell which is enough to accomplish what we want.
Dash
On debian-like systems, dash is the default shell (/bin/sh). The PPID approach works for dash but with yet another twist:
$ echo $$; ( dash -c 'echo $PPID' )
5791
5791
$ echo $$; ( : ; dash -c 'echo $PPID' )
5791
5791
$ echo $$; ( dash -c 'echo $PPID'; : )
5791
20961
With dash, placing the : command before the command is not sufficient but placing it after is.
POSIX
PPID is included in the POSIX specification.
Portability
mklement0 reports that the following works as is with bash, dash, and zsh but not ksh:
echo $$; (sh -c 'echo $PPID' && :)