Does Shiva and Parvati live in Mount Kailasa in invisible form ? As He is not seen there by normal people.
The Kailasha is a spiritual realm. Though it's Present on earth it's dimension or āyām is different. Same as Vrindavan etc are also present in goloka and earth and saket (ayodha). So, though they have same name but dimension are different. But the significance of that place is same as that of the spiritual one.
Sri Shiva Purana 5:19.:
33. O great sage, there are worlds above the universe. Please listen. I shall explain them clearly but briefly.
34. The world Vaikuṇṭha is beyond the world of Brahmā. It is here that Viṣṇu is stationed and it shines with great brilliance.
35. Above it is the extremely wonderful world Kaumāra. The general Kārttikeya, the brilliant son of Śiva shines here.
36. Beyond that shines the excessively divine world of Umā, where Śakti-Śivā, the mother of the three deities shines.
37. She is the Prakṛti, greater than the greatest, with the attributes of Rajas, Sattva and Tamas. But the goddess Śivā herself is devoid of attributes, free from aberrations.
38-39. The eternal Śivaloka is above that. Here lord Śiva, the great Brahman, shines. He is indestructible, excessively divine and endowed with great brilliance. He is the lord of all, greater than the three attributes. He is the progenitor of the three deities.
40. There are no worlds above it. The Goloka is near it. Mother cows named Suśīlā are there. They are favourites of Śiva.
41. The protector of that world is Kṛṣṇa. He is established there at the behest of Śiva by Śiva himself who moves about as he pleases due to his power.
42. O Vyāsa, Śiva’s region is wonderful and beautiful. It has no support. It shines with different objects. It cannot be specifically described.
43. The presiding deity of that region is Śiva. He is the crest-jewel of all the gods. He is worthy of being served by Viṣṇu, Brahmā and Śiva (Rudra). He is the unsullied great soul.
As i have mentioned, in this answer of mine Kailasha is beyond places of many celestial beings, gandharvas, Vishnupada, etc. So, it's clear that it's a celestial region.
Yoga Kundalini Upanishad.:
ज्वलनाघातपवनाघातोरून्निद्रितोऽहिराट् । ब्रह्मग्रन्थिं ततो भित्त्वा विष्णुग्रन्थिं भिनत्त्यतः॥८५॥ रुद्रग्रन्थिं च भित्त्वैव कमलानि भिनत्ति षट् । सहस्रकमले शक्तिः शिवेन सह मोदते॥८६॥ सैवावस्था परा ज्ञेया सैव निर्वृतिकारिणी॥८७॥
1.85-87. From the blowing of Vayu and Agni, the chief (Kundalini) pierces open the Brahmagranthi and then Vishnugranthi. Then it pierces Rudragranthi, after that, (all) the six lotuses (or plexuses). Then Sakti is happy with Shiva in Sahasrara Kamala (1000 lotuses seat or pineal gland). This should be known as the highest Avastha (state) and it alone is the giver of final beatitude (Moksha).
Sahasrara Chakra is also called the city of Brahman.:
“taM chedbrUyuryadidamasminbrahmapure daharaM puNDarIkaM veshma daharo.asminnantarAkAshaH kiM tadatra vidyate yadanveShTavyaM yadvAva vijij~nAsitavyamiti sa brUyAt.h | (Chandogya Upanishad VIII-I-2). yAvAnvA ayamAkAshastAvAneSho.antarhR^idaya akAsha ubhe asmindyAvApR^ithivI antareva samAhiteubhAvagnishcha vAyushcha sUryAchandramasAvubhau vidyunnakShatrANi yachchAsyehAsti yachcha nAsti sarvaM tadasminsamAhitamiti |” (Chandogya Upanishad VIII-I-3).
“If the disciples should say to him, ‘In this city of Brahman in which is a small mansion in the shape of a lotus and in the small inner Akasa within – what is it that lies there which should be sought, which one should desire to understand ?’ – He should say in reply, ‘As large indeed as is this Akasa, so large is that Akasa in the heart. Within it, indeed, are contained both heaven and earth, both fire and air, both the sun and the moon, lightning and the stars. Whatever there is of him in this world and whatever is not, all that is contained within it’”.
No, surgeon has ever found Sahasrara Chakra while doing Brain Surgery or Sushuma Nādi where the whole universe is located in Heart Surgery. Or Sutle body during Post Mortem, etc nor they ever will. Because these are all spiritual, only an Awakened Yogi can see them. Same with Bhagawan also. Though they are Sarva Vyapi, present in every atom of the universe, the eyes of flesh can't see them only through division wisdom eyes we can see them.
I hope this clarifies all your queries. Prd..