Key issues appear to be not so much house prices per se as the affordability of getting onto the housing ladder. Unemployment also appears to be a factor - the two 'peaks' at either end of the ONS graph also correspond to higher percentages of young people who are unemployed (people living with their parents were twice as likely to be unemployed). Intriguingly, the West Midlands had the highest percentage of age English region, with 29% of 20-34s living with their parents. Younger adult men are also more likely to be living with their parents than younger adult women.
The numbers decrease with age (at least one paper's headline reads 'a quarter of 34-year olds living with their parents' which is in fact incorrect) - ONS data suggests that amongst 20 year-olds, 65% of men and 58% of women were living with their parents, but this falls to 8% of men and 3% of women by the age of 34.
All this is very interesting - a few generations ago living with one's parents until marriage was a fairly normal pattern for many young adults, and newly-weds often started married life living at one or other parents' houses until enough had been saved for a mortgage. As with the fall and rise in the age of marriage and childbirth, this most recent set of data suggests once again that some of the classic/popular markers of full/independent adulthood are being pushed later into one's 20s and 30s as 'young adulthood' is elongated.