Mostly by statistical analyses of the genomes of
living people. You don't need any archaeological DNA at all. And it wouldn't help as DNA degrades very fast. We can't extract readable DNA from anything older than about 1.5 million years at the best of conditions. There are methods to deduce some functions of older DNA by analyzing peptides produced by the DNA but they are vague.
The ages of mitochondrial Eve and Y-chromosomal Adam are determined by the diversity and the known mutation rate. The mutation rate is not very precise so the estimated age of Eve and Adam come with a wide margin of error.
It has very little to do with archaeology so countering a post about genomics with one about archaeology, like
@SDavis did raises the question if they know what they are talking about.
The "Neanderthal era" overlaps widely with the "sapiens era".
Btw.: the hybridization of sapiens with Neanderthalensis is a factor in the determination of the age of the most recent common ancestors and here we needed ancient DNA as no living Neanderthals are known.