I've been reflecting on this one, as I set a PB for running a marathon last October (the 11th I'd run), on trails, some muddy, beating a best that was set on the road, at age 53.
There's two things I'd consider. Firstly, my training regime changed - I was running at least a half marathon on trails every weekend, which was the most disciplined I'd ever been at running - which clearly had an adaptation benefit. But a big part of this was simply having a better running technique - being better balanced on rough ground and more confident at sustaining speed over awkward terrain was a big part of time saving. I think that can be applied to cycling as well - looking at cornering, descending, and keeping momentum through those 5 - 20m dips that are so common on British roads. That's not about more power but applying what power you have better.
Secondly, my first half marathon was run at the tender age of 47, so I didn't have any youthful bests to get in the way of setting a new PB. I've not been able to achieve the same in cycling where my TT PBs were set in my mid 40s, about 10 years after taking up cycling seriously. That's because I'd got close to my potential (or at least what could practically be achieved whilst remaining employed and married) at that point and since then the gradual decline of age (particularly maximum heart rate) got in the way of getting fast. I'm just trying to slow down as little as possible.