Drain the cooked potatoes well and place in a bowl. Set aside until they are cool enough to handle.
Cut any really large ones in smaller pieces, just right for picking up with a fork.
Pour over the olive oil and vinegar and toss the potatoes gently to coat well. It really depends on how many potatoes you have how much oil and vinegar you will need, but it is definitely okay to have some oil and vinegar left in the bottom of the bowl for the potatoes to marinate in. If you have a lot of potatoes, double the oil and vinegar quantities.
Cool to room temperature and pop in the fridge. If you happen to open the fridge for any reason, just give them a gentle toss.
In the morning or after an hour or so, take the potatoes from the fridge and drain off any excess oil and vinegar. You still want the potatoes covered in the oil and vinegar mixture, but you don’t want any left in the bottom of the bowl.
Transfer the potatoes in a large mixing bowl.
Finely chop up the salad vegetables you are using.
Add the vegetables to the potatoes along celery salt, pepper and parsley flakes.
Gradually add the mayonnaise to coat the potatoes and vegetables. You want the potatoes well coated but you don’t want to overdress the salad as it already has a nice coating of the marinade.
Stir the salad very gently to avoid breaking up the potatoes too much. When you are happy with the consistency of the salad, taste it and adjust the seasoning if necessary.
Cover and chill until ready to serve. Garnish with more parsley before serving if desired.
The salad will keep for a good 24 hours from this point.