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the stage at which basically it doesn't need to be any more complicated than just running more
Pretty much this. If you're an occasional runner, then just increasing the frequency of runs will get you fitter, along with very gradually increasing the distance of (some of) your runs. Add in a 5k mid-week, but take it at an easier pace than parkrun. If you can do 2/3 runs a week at an easy pace, this will help you to increase the distance of one of your runs and still recover enough for the next session.
"Pace band" ready for pacing duties at Ladywell 10,000.
Honestly it sounds like you're at a similar stage to me, the stage at which basically it doesn't need to be any more complicated than just running more. Maybe not a full 5k more, that's going to double your weekly distance, or if you're want to do the distance do it gently so you're not too fatigued (prone to injury) going into the parkrun