1. Bring a large pot of salted water to a boil. Add potatoes and cook until tender but still firm, about 15 minutes. Drain, cool and chop.
2. In a large bowl, combine the potatoes, mayonnaise, sweet pickle relish, sugar, onion, mustard, vinegar, celery, pimentos, carrot, parsley, pepper and salt. Mix well, chill and serve.
My Note: I used to throw out potato salad after a few days but I never have any leftover now. I never add the carrot or pimento or parsley and only add the celery when I have it. I have though, taken to mixing this dressing in a small bowl beforehand. I have always had a bit of a time, either making my potato cubes too hard, undercooked, or too mushy and they mash up too much when trying to blend in the dressing ingredients. Pre-mixing the dressing in the bowl, I can cook my potato cubes fairly soft and then gently fold in the pre-made dressing only to toss the onions, celery and potato cubes and eggs togehther and just coat with the dressing. A perfect chunky potato salad!
For my family's personal preference I used 5 hard boiled eggs, mashing up the yolks into the mayo mixture, for a eggy tasting dressing. I also added another teaspoon of grey poupon mustard, left out carrots.