Painting a door frame

I understand that the correct way to paint a door frame is the top part first and then the 2 side parts. Does this change if you are using water based paint?

My thinking is that if I do the top part and then one of the side parts by the time I get to the 2nd side part the top has dried enough so that I could have issues with dragging.

Obviously I should learn to paint quicker so the possibility of this happening is reduced but is there any reason why I shouldn't start painting from the bottom of one of the side parts, over the top and then down the other side part?

Personally, with water based paints, I always paint the inside of the frame first - the jambs and doorstop - and then do the outer section - the archtrave. This means that you can wipe off any paint that finds its way onto the face of architrave before it dries and then when you come to paint the architrave itself, you have a nice even surface which you can do in one hit.
As you say, it's not so essential with oil paints but, with water, you have to work quickly to avoid a poor finish. Using a slightly dampened, synthetic bristle brush, will give you the best results.

Thanks, that's makes sense. The door will be painted last for this reason. But is there any logical reason not to paint the frame up, across and down (whilst I work on my painting speed!!)? Would the finish look different on the sides as I would be laying off upwards on one and downwards on another?

I think starting at the bottom is a recipe for disaster, I would imagine you will be more prone to runs, and it will be hard to see what it looks loke as you're painting.

If you neatly paint into the corner of the frame, you shouldn't have too many issues.