Guilt: "I will leave regardless of your involvement. Failing to render assistance would be neglectful."
This... probably won't work and sounds suspicious.
Social Standing: "That boy said that I was plain-looking! ..."
This will probably result in some line about ignoring such people instead of money.
Lie: "The college requires a fee for enrolment..."
This might actually work, but we had better have everything planned out about whatever we say we need the money for.
(almost) Honesty: "I found an amulet that seems to resonate with my magical properties. I simply must have it!"
Just 'cause: "Well I was in the fair the other day... The food is obscene, but I couldn't attend the fair without eating some... I heard that there were alcoholic mana potions, not that I would have anything to do with such things... I saw the most perfect little amulet. The merchant thought that it was dwarven but it obviously wasn't. I simply must have it, can I please have 100 gold, pleeeeeease! ..."
These two might work, but only if it's easily affordable for them.
Misfortune: "Some ruffian just accosted me in the street. They took all of my monetary resources!"
Our parents probably knows that we don't have much in the way of money, so this won't get us much.
My other car: "I am going into business making mana potions, I need some funds for the initial expenses."
This could work, if it's not questioned to much.
Fancy Dress: "The social season is approaching and I need a new outfit for formal occasions."
Do elves have formal occasions?
Moo Moo Cow Moo: "The fortune-teller said that they would curse me if I didn't pay for their broken stall."
I have nothing to say about this.
For all of these, we have to consider how well it fits with out parents conception of our personality, any sudden changes will be suspicious.