In Shiva Puran according to some sources it is narrated that Parvati and Shiva has had carnal relation through which Hanuman God was born in the form of Vanara and after that, not wishing to carrying his foetus to her womb, Goddess Parvati offered Him to Vayu Wind God. This one version does not find compatible narrations in some other puranas and spiritual texts related to the life of God Hanuman. In the Hanuman Chalisa it is understandable - among the qualities of the God in the first 6 shlokas - that He is the son of the Wind God (Vayu) and a partial aavatar of God Shiva. I am asking if what I have found about the history of the birth of God Hanuman is correct and eventually to receive some more source of knowledge about this topic.
Sri Shiva Purana 3.20.:
O sage, henceforth listen lovingly to the story of Hanūmat, how Śiva performed excellent sports in that form.
Out of love, Lord Śiva rendered great help to Rāma. O brahmin, listen to the whole of that story which is pleasing to all.
Once lord Śiva of wonderful sports saw Viṣṇu in the form of an enchantress (mohinī) with glittering qualities.
As if hit by the arrows of Cupid, Śiva let fall his semen dislodged from its seat, for Rāma’s work.
Eagerly urged by him mentally for Rāma’s work, the seven celestial sages retained that semen in a leaf.
For Rāma’s work that semen was poured through the ears of Añjanī, the daughter of Gautama, by those sages.
In due course Śiva was born of it in the form of a monkey named Hanūmat. He had great strength and exploit.
So, Actually, Shiva Purana itself not mentioned Hauman's birth between Shiva-Parvati's union. However, it is mentioned that Lord Hanuman's birth after Bhagawan Shiva saw Lord Vishnu's Mohini incarnation, which was again the form of Mata Tripura Sundari herself.
Sri Brahmanda Purana.: Lalitopakhyana.: 2:04 & 8.:
“Meanwhile, Vishnu began to meditate on Sri Lalita Mahatripurasundari, who was his very own Self.”
“By completely identifying himself with Maheshwari, Vishnu manifested himself in her form (as Vishwa Mohini).”
I hope this clarifies all your queries. Prd..