According to this article, DNA has a half life of 512 years
Wikipedia claims that there are seeds planted as old as 31000 years
Doing the basic math $100 / (2^{30000/512})$ we get that only about $2.2989191*10^{-16}%$ of the DNA would be left
So how do these plants still grow when replanted?
That half-life was for temperatures of 13.1ºC. In more ideal conditions, such as for temperature of −5 ºC or drier conditions, the half-life would be longer. The Wikpedia article in fact states that the seeds that were 31,000 years old were frozen under permafrost.
The original article estimates that the half-life at −5 ºC (for a 30 bp fragment of mtDNA from bone) would be 158,000 years. I'm not sure how well this data can be extrapolated to seeds.
The source paper for the article is this one: The half-life of DNA in bone: measuring decay kinetics in 158 dated fossils (available for free). The crux of their methodology is given in the abstract "By analysing mitochondrial DNA (mtDNA) from 158 radiocarbon-dated bones of the extinct New Zealand moa..."
So the article you linked is talking about DNA survival after cell death in fossilised bones. Seeds have not undergone cell death; they're still a small collection of living cells designed to survive in a state of near inactivity until the conditions are suitable for them to germinate. It's therefore not at all clear that we should expect the results from this experiment to generalise to seeds. Doubly so given that, as pointed out by Jonathan, the fossils were effectively held at around 13.1ºC whereas the seeds in the study you mention were found in the permafrost.
Finally, the seeds weren't viable on their own. It was only with significant intervention that scientists were able to get the embryos contained within the seeds to grow.
