Parashurama engages in a duel with Rama after Rama broke the Shiva dhanush (bow). But why couldn't Parashurama, an avatar of Lord Vishnu, recognize Rama who is also another avatar of Lord Vishnu, even before engaging in a duel with Rama?
Although Lord Parashuram is still living incarnation of Lord Vishnu. He is not full incarnation as Rama and Krishna..
To be called Bhagwan (God) one should possess these six qualities as per Vishnu Purana
aiśvaryasya samagrasya dharmasya yaśasariśrayaḥ
jñānavairāgyayoścaiva ṣaṇṇāṃ bhaga itīraṇā [VP - 6.5.74]
Complete splendor, virtue, glory, opulence, knowledge, dispassion - these six are known as bhaga.
But as per Padma Purana out of 10 incarnations only 3 incarnations (Nrisimha Rama, Krishna) possessed all these six qualities.
Padma Purana Uttarkhanda Chapter 229
40-42. Thus the state of the inner-controllership of self consists in his being the innermost soul. Matsya, Kurma,Varaha, Nrisimha, Vamana, Rama, Parasurama, Krishna, Buddha and Kalki are the ten incarnations of Brahman, the highest soul. The group of six qualities is said to exist in Nrisimha, Rama and Krishna.
Hence as he wasn't full incarnation of Lord Vishnu so he couldn't recognize Lord Rama in human form at first time.. But this is only just a part of Leela of Lord Vishnu as it is not necessary that to recognize Lord one have to become full incarnation as just a devotee can also recognize Lord.