Your cost is the donor's cost, not the gift value
A gift does not reset the cost to zero or to what the property was worth on the day it was gifted. The law carries the donor's cost across to you (Section 49(1)): whatever the person who gifted you the property originally paid for it becomes your cost of acquisition when you sell. So a flat a parent bought for a few lakh is treated as having cost you those few lakh, not nothing.
Where the donor acquired the property before 1 April 2001, you get the same relief any old owner would: you may substitute its fair-market value as on 1 April 2001, fixed by a registered valuer and capped at the stamp-duty value of that date, as the cost (Section 55). Because that 2001 value is far higher than an old purchase price, it is usually the figure that most reduces the gain on a long-held gifted property.
The donor's holding period counts, so the gain is long-term
For working out whether the gain is long-term or short-term, you do not start the clock on the day the gift was made. Your holding period includes the period the donor held the property (Section 2(42A)). So if a parent held the flat for fifteen years before gifting it to you last year, the property is long-term in your hands, not short-term.
That matters because long-term property gains for an NRI are taxed at the flat 12.5% rate with no indexation for sales on or after 23 July 2024 (Section 112), whereas a short-term gain would be taxed at your slab rate, which is usually much higher. The carried holding period is what keeps a recently-gifted but long-held family property on the lower long-term footing.
Receiving the gift, versus selling it, are two different events
People worry that the gift itself was taxed and will now be taxed again on sale. In most family cases it was not taxed at all: a gift of property from a relative, the defined list that covers parents, spouse, siblings and lineal ascendants and descendants, is exempt on receipt. A gift from a non-relative worth more than fifty thousand rupees is taxable on receipt under Section 56(2)(x).
Where the gift was taxed on receipt, the law makes sure it is not taxed twice: the value that was charged becomes your cost when you later sell (Section 49(4)), so only the appreciation after that is taxed as gain. So the two events are handled cleanly, a relative gift is tax-free on receipt and carries the donor's cost, and a taxed non-relative gift carries the taxed value as its cost. Either way the pre-gift value is not taxed a second time.