So the concept is to allocate into various assets not based on nominal amount, but rather based on risk which you can do using some modicum of leverage. At lot of the happiness embedded in the strategy takes advantage of the low-to-negative correlation between bonds and risk assets. If I had to guess, there is probably about 500 billion invested in that strategy right now, with another few hundred doing it in a discretionary form.
Their exact actions depend, obviously, on the exact definition of the "risk parity" and the mandate of the fund. In a classical form, the risk is defined as volatility so if bonds became more volatile you'd had reallocated away from bonds, most likely into equity. So you can easily get a yo-yo type effect where they reallocate into equities, equities sell-off, they reallocate into bonds etc.
