How to mark the text between the parentheses in Emacs?

expand-region (which is bound to C-=) works great.

Try the key sequence C-M-u C-M-SPC (i.e., while holding the Control and Meta keys, press u and Space in sequence), which executes the commands backward-up-sexp and mark-sexp.

Edit:

I made a mistake: backward-up-sexp doesn't exist in standard Emacs. I wrote it exactly because of the problem mentioned in lkahtz's comment, that the existing function backward-up-list won't work when point is between double quotes.

(defun backward-up-sexp (arg)
  (interactive "p")
  (let ((ppss (syntax-ppss)))
    (cond ((elt ppss 3)
           (goto-char (elt ppss 8))
           (backward-up-sexp (1- arg)))
          ((backward-up-list arg)))))

(global-set-key [remap backward-up-list] 'backward-up-sexp)

Xah Lee has an emacs-lisp function which achieves this called xah-select-text-in-quote. It is available from his website:

Select text between the nearest left and right delimiters. Delimiters here includes the following chars: \"<>(){}[]“”‘’‹›«»「」『』【】〖〗《》〈〉〔〕（）. This command does not properly deal with nested brackets.