A Vaishnava-based answer ahead…
No. Lord Parashurama is an incarnation of Lord Vishnu. It is very important to know which are Lord Vishnu’s avatars and which are not. Out of the infinite forms of Lord Vishnu, which are non-different from each other, Parashurama is also a form. Here are some proofs of Lord Parashurama being an incarnation of Lord Vishnu.
यमाहुर्वासुदेवांशं हैहयानां कुलान्तकम् ।
त्रि:सप्तकृत्वो य इमां चक्रे नि:क्षत्रियां महीम् ॥ १४ ॥
Learned scholars accept this Paraśurāma as the celebrated incarnation of Vāsudeva who annihilated the dynasty of Kārtavīrya. Paraśurāma killed all the kṣatriyas on earth twenty-one times.
[SrimadBhagavatam, 9.15.14]
evaḿ bhṛguṣu viśvātmā
bhagavān harir īśvaraḥ
avatīrya paraḿ bhāraḿ
bhuvo 'han bahuśo nṛpān
In this way the supreme soul, the Supreme Personality of Godhead, the Lord and the supreme controller, descended as an incarnation in the Bhṛgu dynasty and released the universe from the burden of undesirable kings by killing them many times.
[SrimadBhagavatam 9.16.27]
जामदग्न्योऽपि भगवान् राम: कमललोचन: ।
आगामिन्यन्तरे राजन् वर्तयिष्यति वै बृहत् ॥ २५ ॥
My dear King Parīkṣit, in the next manvantara the lotus-eyed Personality of Godhead Lord Paraśurāma, the son of Jamadagni, will be a great propounder of Vedic knowledge. In other words, he will be one of the seven sages.
[SrimadBhagavatam, 9.16.25]
Appearing in the forms of a swan, a tortoise, a fish, O foremost of regenerate ones, I shall then display myself as a boar, then as a Man-lion (Nrisingha), then as a dwarf, then as Rama of Bhrigu's race, then as Rama, the son of Dasaratha, then as Krishna the scion of the Sattwata race, and lastly as Kalki. When the auditions in the Vedas disappeared from the world, I brought them back. The Vedas with the auditions in them were re-created by me in the Krita age.
In the Treta age I shall take birth as Rama in the race of Bhrigu, and exterminate the Kshatriyas who will become proud of their strength and possessions.
[Mahabharata, Shanti Parva]
Even SriMadhvacharya in His Mahabharata Tatparya Nirnaya says that, “Matsya, Kurma, Varaha, Narasimha, Vamana, Parashurama, Rama, Krishna, Buddha, Kalki, Hari, Vedavyasa, Aitareya, Kapila Vaasudeva, Dattatreya, Badavalana, Dhanvantari, Mohini, Hayagriva, Yagya, Taapasa, Narayana, Rishabhadeva, etc. are verily Lord Vishnu Himself and there is no difference between Lord Vishnu and His avataras.”
Lord Krishna in the Bhagavad Gita says that He is the one who is worshipped in all the yagyas. There is a yagya by the name Vaishvadeva, which is done for anna shuddhi; the main Bhagavadroopa who is being worshipped is Lord Parashurama roopi Narayana residing within Agnideva. Parashurama is the antaryami roopa or the indwelling form within Agni. Even when we show Naivedya to Bhagavan Vishnu, Bhagavan accepts the Naivedya in His Kurma roopa, smells the Naivedya in Narasimha roopa, and gives it to Lakshmidevi. Then He gives one glance at the Naivedya in His Parashurama roopa, which is then accepted by other devatas as Bhagavat prasaada.
If you have a question regarding the Parashurama and Rama encounter in Treta yuga, you can refer to this answer of mine. In this, I have explained why it happened and all, as per SriMadhvacharya’s Mahabharata Tatparya Nirnaya.
In the Vishnu Sahasranama too, there is a name of Lord Vishnu by the name “सुधन्वाखण्डपरशुर्दारुणो द्रविणप्रदः ।” where Akhanda parashuh means the one possessing an unbroken axe, and we know it is Lord Parashurama only. Plus, He is also considered one of the Dashavatara of Lord Vishnu.
Hope this helps!
Jay SriRam!
ShriKrishnarpanamastu!