Short answer :
It is a common misconception outside the Shâkta circle that Bhagavatî Durgâ is a manifestation of Pârvatî Devî, the wife of Bhagavâna Shankara.
O Brahmin, best of my sons, listen attentively to what I did when the lord Viṣṇu went away. I began a continuous laudatory prayer of the Goddess Durgā, the beloved of Śiva, the creator of the universe, of the nature of Vidyā and Avidyā and identical with the pure supreme Brahman.[Shiva Purâna:2:2:11:2-3]
O celestial sage, on being thus lauded Caṇḍikā, the mystic slumber, appeared before me.[Shiva Purâna:2:211:6]
O sage, seeing her who was Śiva’s Energy herself, directly in front of me, my lofty shoulders bent down with devotion and I eulogised her after due obeisance.[Shiva Purâna:2:2:11:9]
O Goddess, be pleased with me and listen. O omniscient Goddess, I speak out my mind only since you have commanded me thus. O Goddess of devas, Śiva the Yogin, who is your husband and who as Rudra manifested Himself formerly through my forehead has now occupied Kailāsa. The lord of Goblins is performing penance all alone. Since He does not desire a wife He is without a wife. He is free from mental aberrations. O Satī, fascinate Him lest He should cast His glance on another lady. Excepting you none will be able to capture His mind. Hence, you alone should fascinate Śiva through your beauty. O Śivā, being born as Dakṣa’s daughter you should become Rudra’s wife. Just as assuming the physical form of Lakṣmī you delight Viṣṇu, so act similarly to Rudra for the benefit of the universe.[Shiva Purâna:2:2:11:20-25]
Now the identity of the one who is mentioned by the names of Durgâ, Chandikâ, Shivâ is described
A form that goes everywhere, that has all forms, that sees all, that is the cause of all, that should be respected by all, that is at the beginning of all, that bestows everything, and that sanctifies everything should be created (So it wished) and hence created that form of Īśvara of pure nature. The original Being without a second, with neither beginning nor end, that illuminates everything, that is in the form of Cit (pure knowledge), that which is termed Supreme Brahman, the all-pervasive and undecaying, vanished, The manifest form of the formless Being is Sadāśiva. Scholars of the ancient and succeeding ages have sung of it as Īśvara. Īśvara though alone, then created the physical form of Śakti from his body. This Śakti did not affect his body in any way. This Śakti is called by various names. Pradhāna, Prakṛti, Māyā, Guṇavatī, Parā, the mother of Buddhi Tattva (The cosmic Intelligence), Vikṛtivarjitā (without modification). That Śakti is Ambikā, Prakṛti and the goddess of all. She is the prime cause and the mother of the three deities. She has eight arms. Her face wears a peculiar splendour, the splendour of a thousand moons. Thousands of stars perpetually sparkle round her face. She is bedecked in various ornaments. She has various weapons. She is capable of various movements. Her eyes beam like a full blown lotus. She has a brilliance which could hardly be conceived. She is the generating cause of all. She sprang up singly as Māyā. In her union she manifested in various forms.[Shiva Purâna:2:1:6:16-24]
The first primordial Śakti of Śiva the great Brahman, who is called Umā, is the great mother of the three worlds. Her two incarnations—Satī and Haimavatī, have been heard, O intelligent Sūta. Please mention her other incarnations too.[Shiva Purâna:5:45:3-4]
The following verses spoken by Shrî Sûta after this within the same chapter describes the killing of Madhu-Kaitabha under the influence of Mahâkâlî. The following chapters then recount the killing of Mahishâsura & Shumbha-Nishumbha. From this & other verses quoted above, it is clear that Bhagavatî Durgâ is NOT a manifestation of Pârvatî Devî.
It is seen that some people are quoting from the Devîbhâgavata Purâna to justify their errored notion.
Vyāsa said :-- O King! Hearing thus,Śaṅkara smiled a little and spoke charming words to the Lotus-born in the following manner :-- O Bibhu! It is You that gave before this boon to Mahiṣa; and therefore it is you that have wrought this mischief. The Dānava(Mahishâsura) has become so strong a hero that he has caused terror to all the Devas even. Now where can we get such a noble woman who becomes able to kill that Dānava, elated with pride. My wife nor your wife ought to go to battle; even if they, the good ladies go, how will they be able to fight? The fortunate wife of Indra, too, is not expert in the art of warfare; where else there is another lady who can kill this demon, blinded with pride. I, therefore, propose this; let us all go today to Viṣṇu and, praising him with hymns, engage him quickly to this cause of the gods. Viṣṇu is foremost amongst the intelligent; therefore it is highly advisable to execute all actions after duly consulting with him. He, by dint of his high intelligence, will find out means and effect our purpose.[Devîbhâgavata Purâna:5:7:48-55]
The emboldened words clearly show that Bhagavatî Durgâ isn't a manifestation of the wife of Bhagavâna Shankara like Bhadrakâlî, the slayer of Dârukâsura mentioned in Linga Purâna. If She had been so, then why Bhagavâna Shankara isn't addressed by names like 'Lord/Beloved of Durgâ/Chandikâ/Kâtyayanî' etc ?
The counterargument which is forwarded is : Then why is Bhagavatî Durgâ is addressed by the names of 'Pârvatî', 'Beloved of Shiva', 'Spouse of Bhava' etc ? The answer lies in the same logic by which names that are specifically used to address Bhagavâna Krishna(like Govinda & Vâsudeva) are also used to address Bhagavâna Vishnu.
One can argue that Bhagavatî Kaushikî emerged out out of the skin of Pârvatî Devî to validate their argument but I would like to tell them:- Both Shiva Purâna & Vishnu Purâna agree that Bhagavâna Kâlâgnirudra manifested out of an angry Prajâpati Brahmâ's forehead but still He is regarded as an expansion of Bhagavâna Shankara not of Paramesthi Brahmâ.
All this arguments justifies the fact that Bhagavatî Durgâ & Devî Pârvatî are the manifestations of Âdyâshakti Mahâmâyâ.