Sometimes it is written in different sites that Parvati slept during the Amarkatha and it is said that she is immortal. What is true?
Yes Devi Uma or Devi Parvati is eternal goddess – She is ever present with Lord Shiva the eternal god - as his consort/companion .
In the commentary of the verse No . 25 from The Kena Upanishad , Adi Shankaracharya wrote that Uma the daughter of Himalaya is ever associated with the Lord (Siva).
स तस्मिन्नेवाकाशे स्त्रियमाजगाम बहुशोभमानामुमाँहैमवतीं ताँहोवाच किमेतद्यक्षमिति ॥ 25 ॥
25 He beheld in that very spot a woman, Uma, very beautiful and of golden hue, daughter of Himavat. He said to her “What is this Great Spirit?”
Shankara’s Commentary: This epithet is very appropriate in the particular context-as if adorned in gold. ‘Himavatim’ may mean ‘the daughter of Himalaya’ and being ever associated with the Lord (Siva) the omniscient, and having approached her, asked: “Who is this Spirit that showed itself and vanished?”
And this verse from Svetasvatara Upanishad page No. 62, declares that Rudra Or Shiva is the eternal god - Brahman.
यो देवानां प्रभवश्रोच्वश्रच् विश्वाधिपो रुद्रो महर्षि ।
हिरण्यगर्भं जनयामानस पूर्वं स नो बुद्धया शुभया संयुनक्तु।।4||
He, the omniscient Rudra, the creator of the gods and the bestower of their powers, the support of the universe, He who, in the beginning, gave birth to Hiranyagarbha−may He endow us with clear intellect!
4 That God is eternal is subject is also implied.
It’s also said in Rudra-Hridaya Upanishad that the Vyakta is Sri Uma, and the Avyakta is Lord Siva i.e Prakriti and Purusha. Although Uma and Shiva are not separable from each other Uma being Prakriti (nature) dies and reborn again due to various causes and reborn again, yet being the eternal part of Supreme Purusha.
Parvati's lifetime is 6 months in the life of Brahma. Here is what Shiva tells Parvati in this excerpt from the Prabhasa Khanda of the Skanda Purana:
In this Varaha Kalpa, O goddess, you became Parvati obtained by Himavan through his penance, when the Chakshusha Manvantara has passed off. O beautiful lady, due to the anger of Daksha you had to be separated from me for the duration of a day of Brahma. Your span of life is the period of his six months.
It says Shiva and Parvati got married "when the Chakshusha Manvantara has passed off", so she was either born close to the end of the Chakshusha Manvantara or the beginning of the Vaivasvata Manvantara. And she'll live for 6 months of Brahma, which means 180 Kalpas. So Parvati's lifetime is 200 times less than Brahma's lifetime.
In any case, before she was Parvati, she was Shiva's first wife Shakti, and she lived in the Pitri Kalpa.