"/r/ + t + vowel" pronunciation in American English

Solution 1:

Your ears have become too accustomed to British English, and you aren't hearing the "r" in party in the dictionary pronunciation, which is obvious to me, an American.

Possibly you don't hear the /r/ in party because it's been absorbed into the /ɑ/ ... it's not an /ɑr/ but an /ɑ˞/, an /r/-colored /ɑ/, where you pronounce the /ɑ/ and the /ɹ/ kind of simultaneously. These are allophones in American English.

The same thing is happening in flirty – there's an r-colored /ɜ/. And here, in fact, there's no /r/ in the IPA transcription /ˈflɝː.t̬i/ because they use the symbol /ɝ/, which stands for an r-colored /ɜ/. Since most Americans use an r-colored vowel in flirty and pertain, but not in party, you generally only see two r-colored phoneme symbols in dictionary transcriptions, /ɝ/ and /ɚ/ (and most dictionaries use /ɜr/ and /ər/, which is less confusing and thus probably a better decision).

Let me say that I use the r-colored vowels /ɝ/, /ɚ/, /ɔ˞/, /ɑ˞/, and /ʊ˞/, but say /ɪr/ and /ɛr/. However, which vowels are r-colored, and which have an /r/ after them varies depending on the speaker.