She strolled slowly through the park, trying to show off her new hairstyle.
Literal
She [topic-は] new hairstyle [object-を] show-off-try-to [doing-して] park-inside [through-を] slowly walked.
見せびらかす (みせびらかす) means 'to show off, to flaunt.' The ~ようとして uses the volitional form + とする ('to try to') in te-form to indicate her ongoing intent while walking. ゆっくり ('slowly') is an adverb that implies a deliberate, leisurely pace — she's walking to be seen. ヘアスタイル is an English loanword fully integrated into Japanese.