Are Lalita and Goddess Parvati the same?

 Yes, Goddess Parvati is Raj Rajeshwari Bhagwati Lalita Tripurasundari.

Maa Tripurasundari is more transcendental and when she incarnates in this realm as the daughter of King Himavan, she is known as Parvati.

Tripurasundari is the most beautiful among the three realms. Basically she is the Ultimate source of beauty, the entire beauty, whatever is beautiful, it's her manifestation.

She is also known as Kamakshi, she is the consort of Kameshwar Mahadev but Mahadev also resides with him in multiple forms like Mahesh, Rudra, Bhairav, Ishan, not just as husband (Kameshwar) but also in these forms to serve her out of love.

She sits on Sadashiva, another auspicious form of Mahadev. This symbolizes the active, flowing energy, i.e., Shakti over the static, unshakable consciousness, i.e., Shiva. Mahadev is the Aadhar (Base) and Mahadevi is the projection.

Jai Maa Lalita Tripurasundari 🙏 Har Har Mahadev 🙏

To which goddess does Sri Chakra belong? Is it Sridevi Mahalakshmi (Vishnupatni) or Lalita Tripurasundari (Shivpatni)?

 The Adhiṣṭhātrī Devī of Śrī Cakra is Devī Mahālakşmi. She is Kāmeśvara Maheśvarī, Supreme Consort of Kāmeśvara Śiva. She is the empress of Maṇidvīpa. She is Mahātripurā Sundarī Lalitā.

The Brahmāṇḍa Mahāpurāṇa, which sings the glory of Parāśakti Lalitā, clearly states that Śrī Cakra is her Yantra.

मन्त्रं श्रीचक्रगेवास्याः सेयं हि त्रिपुरांबिका ।
सैषैव हि महालक्ष्मीः स्फुरच्चैवात्मनः पुरा ॥ ३ ॥

3. Śrī Cakra alone is her Mantra (Yantra). She is Tripurāmbikā and she is Mahālakṣmī herself. Formerly, the Cakra was seen coming out of Ātman throbbing and sparkling.

~ Brahmāṇḍa Mahāpurāṇa, Upasaṃhārapāda, Lalitopākhyāna, Chapter 41

त्रिमूर्तिजननीमंबां दृष्ट्वा श्रीचक्ररूपिणीम् ।

119. She who is the mother of the Trimūrtis (Brahmā, Viṣṇu and Śiva) and who is present there in the form of Śrīcakra.

~ Brahmāṇḍa Mahāpurāṇa, Upasaṃhārapāda, Lalitopākhyāna, Chapter 40

कामाक्ष्यैव महालक्ष्मीश्चक्रं श्रीचक्रमेव हि।

15. Kāmākşi alone is Mahālakşmi. The Cakra is Śricakra itself.

~ Brahmāṇḍa Mahāpurāṇa, Upasaṃhāra-pāda, Lalitopākhyāna, chapter 41

तस्या मन्त्रमिति प्रोक्तं श्रीचक्रं चक्रूषणम् ।
नवावरणमीशानी श्रीपरस्याधिदैवतम् ॥ ४ ॥

4. Her Mantra also has been mentioned. So also the Śrī Cakra which has nine outer coverings and the mystic circles as its ornaments. (You have mentioned that) Īśānī is the presiding deity of Śrīpura.

~ Brahmāṇḍa Mahāpurāṇa, Upasaṃhārapāda, Lalitopākhyāna, Chapter 39

🪷 Devī Lalitā Tripurā Sundari 🪷 (PC: saitharunvs)

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Bhāvana Upaniṣad depicts the human body as a Śrī Yantrā (Śrī Cakra), associating each body part with a specific Cakra and in the Śrī Cakra, Devī Lalitā is the supreme divinity.

नवशक्तिरूपं श्रीचक्रम् ।

It is the Śrī Cakra in the guise of the nine wheels.

सदानन्दपूर्ण स्वात्मेव परदेवता ललिता ।

The supreme divinity, Lalita, is one’s own blissful Self.

~ Bhāvana Upaniṣad, Atharva Veda

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One of the Śākta Āgamas, the Vāmakeśvara Tantra, thoroughly explains the worship of the Śrī Cakra. It enumerates the nine Āvaraṇas, lists the Devīs in each Āvaraṇa (Āvaraṇa Devatās), the Cakreśvarī of each Āvaraṇa, meditation on the Bindu of the Cakra, and other details concerning the Śrī Cakra, Mudrās, and Śrī Vidyā. And Vāmakeśvara Tantra says at the Bindu resides Śrī Mahābhaṭṭārike Parāpara Rahasya Yoginī Lalitāmbikā and she move within Śrī Cakra.

त्वां गौरीं त्रिपुरां परात्परकलां श्रीचक्रसञ्चारिणीम्

You are Gaurī, Tripurā, the supreme transcendental energy, who moves within the Śrī Cakra.

~ Vāmkeśvara Tantra

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In the Tripurā Rahasya, Devī herself affirms that the Yantra representing her is the Śrī Cakra.

श्रीविद्योत्यहमाख्याता श्रीपुरं मे पुरं भवेत्।
श्री चक्रं मे भवेच्चक्रं श्रीक्रमः स्यान्मममक्रमः॥ ४६॥

46. Henceforth, I will be renowned as "Śrī Vidyā." My city will be famously known as "Śrīpura". The Cakra related to me will be known as "Śrī Cakra", and the Krama associated with me will be known as "Śrī Krama".

~ Tripurā Rahasya, Māhātmya Khaṇḍa, Chapter 53

🪷 Devī Lalitā and Kāmeśvara Śiva surrounded by the Nithyā Devīs 🪷(PC: swaroop_621)

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While commenting on the name “Śrī Cakra Rājanilayā”, Śrī Bhāskararāya Makhin in his Lalitā Sahasranāma Bhāṣya mentions that Śiva and Devī reside in Śrī Cakra.

श्रीचक्रराजं बिन्दुत्रिकोणादिरूपं निलयो वासस्थानं यस्याः। तदुक्तं'श्रीचक्रं शिवयोर्वपु'रिति। शरीरे यथा जीवस्याव स्थानं तथा श्रीचक्रे शिवयोरिति तदर्थः ।

Śrī Cakra, consisting of Bindu, triangle, etc. It is said, "Śrī Cakra is the body of Śiva and Devī." The meaning is just as the Jiva resides in the body so they reside in the Śrī Cakra.

~ Saubhāgya Bhāskara (Commentry on Lalită Sahasranāma) by Bhāskararāya Makhin

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The Stōtras praising Bhagavatī also states that Devī Lalitā is Śrī Cakra Vāsinī.

श्रीचक्रस्थां शाश्वतैश्वर्यदात्रीं पौण्ड्रं चापं पुष्पबाणान् दधानाम् ।
बन्धूकाभां भावयामि त्रिनेत्रां तामग्निवर्णां तपसा ज्वलन्तीम् ॥ ३९ ॥

I bow to that Devī who shines with penance; who resides in Śrī Cakra; who possess inperishable wealth and who bears sugarcane bow and flowry arrows and has three eyes of reddish hue.

~ Śrī Ādi Śaṅkarācārya Viracita Tripurasundarī Vēda Pāda Stotram

श्रीविद्यां शिववामभागनिलयां ह्रीङ्कारमन्त्रोज्ज्वलां श्रीचक्राङ्कितबिन्दुमध्यवसतिं श्रीमत्सभानायिकाम् ।

Devī Mīnākṣi, Who is Śrī Vidhyā shining with the splendour of Hrīṅkāra mantras, where she is residing as the left-half of Śiva,
Within the bindu, at the center of the Śrī Cakra; and presiding over the assembly of deities in the Śrī Cakra.

~ Śrī Ādi Śaṅkarācārya Viracita Mīnākṣi Pañcaratnam

कुण्डल त्रिविधकोण मण्डल विहार षड्दल समुल्लसत्।

Devī Lalitā, Who shines in the six petals, surrounding the triangle in the middle of the (Śrī Cakra)

~ Śrī Ādi Śaṅkarācārya Viracita Navaratna Mālikā

🪷 Śrī Cakra Rāja 🪷 (PC: Google)

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Note: The Cakra or Yantrā of Devī Lakṣmī is the Lakṣmī Yantrā, which is also known as the Śrī Yantrā since another name for Devī Lakṣmī is Śrī. However, this does not make her the Adhiṣṭhātrī Devī of the Śrī Cakra Rāja. Devī Lakṣmī resides in the Lakṣmī Yantrā, and the methods of worshipping her Yantrā, including the Āvaraṇa Devatās, Dhyāna, and Nyāsa, are entirely different. Thus, the Lakṣmī Yantrā and the Śrī Yantrā are distinct from each other.

🪷Śrīmātrē Namaḥ🪷