。
She managed at last to push her way through the crowd in the park.
Literal
[topic-は] with-great-difficulty park [genitive-の] crowd [genitive-の] inside [path-を] passed-through.
やっとのことで ('with great difficulty, just barely') is an idiomatic intensifier of やっと ('finally') — adding のことで spotlights how arduous the achievement was. 通り抜ける ('to pass through, go through') is a compound: 通る ('pass') + 抜ける ('come out the other side') — together emphasizing the complete traversal. 群集の中を marks the crowd as the path traversed (path-を), with 中 ('inside') giving 'through the middle of.'