In this video, Felix Otto says that the correct way to count dimensions for parabolic equations is $2+\text{number of space dimensions}$. He said nothing about this. In the accompanying notes it is stated, again with no explanation.

When I searched the literature the only place I could find any reference is again more SPDE/regularity structures people. Here it is stated with little explanation. (Page 10)

Other PDE people usually say that time counts as one dimension. Why in SPDE/regularity structures does time "count for" 2 dimensions?

Does it have something to do with complex spacetime and Wick rotation? If we consider time complex it is 2D. However in all discussed SPDEs, time is strictly real.

Edit: I thought about it some more. Does it have to do with parabolic scaling? I.e. we scale $u(t,x_1,...x_n)$ as $u(c^2 t, c x_1,...,c x_n)$. The $t$ term picks up $c^2$ so it "counts twice"?