Drop the false shame. For parents you are the dearest person in the world, no amount of money will be more expensive for them than you. You can experience uncomfortable taking up with friends or colleagues, but parents can trust completely and without hesitation to ask for the right amount.
Veil purpose. If the parents have views on life that are different from your own, you may have a bit to fool them. For example, you want to purchase a new smartphone, and elderly parents against all these newfangled incomprehensible technical products. In this case you can say that you want to purchase a new kitchen utensil that will help you to eat better. Parents will be happy to give you money for useful from their point of view thing. Of course, to deceive parents can only benefit, never ask them money for inappropriate purposes.
Show their best side. If you are underage and live with your parents, you should be a good child for them, try to learn to their maximum potential, help parents, heed their advice, share with them their experiences and then the money won't even have to ask the parents themselves will give them to you if needed.
Maintain relationships with parents, call them often, be more responsive to their problems, just love them. Feeling your warmth and care, they will always be happy to help you. If you call them once in two years only to ask for money, you risk to run into a cold "no."
Create a good reputation in the eyes of parents. No matter how many years you may be, parents always nice to know that their child does well, successfully moving up the career ladder, do not drink or smoke, has a healthy family. Give money to a man with a reputation for drunks and rowdy does not want one, even his own parents.