[3.13] gh-122581: Avoid data races when collecting parser statistics (GH-122694) (#122733)

gh-122581: Avoid data races when collecting parser statistics (GH-122694)
(cherry picked from commit ce0d66c8d238c9676c6ecd3f04294a3299e07f74)

Co-authored-by: Lysandros Nikolaou <lisandrosnik@gmail.com>