I'm currently working as a teacher in a Chinese school. My employer is not the school, but a placement agency.

He's not very useful - he doesn't speak any English and he's a bit of a prick (my flatmate with workable Chinese translates for me). Also, he hasn't yet given me a contract (I've been at the school since about September.) From my calculations he's ripping me off in a big way, too. (The classes with me bring in 30k a month, but I get about 7200 - I guess he and the school split the rest.)

I asked the school if I could work directly with them and skip my boss, next term and they said they would need to provide "fa piao" (tax receipts) to the parents who pay for my lessons. I asked what kind of receipts they were and they showed me one.

It was the same kind you get at Pizza Hut with the scratch and win panels, and it had the stamp of my boss's "company" on them.

Where can I obtain these and if I explain my situation and income to the distributor (some tax dept. office, I presume) and we work out my tax code, will me paying for a book of tickets suffice as having paid my individual income tax?

(I am a New Zealand/Irish national who has been here since May 1st 2004 and I don't intend to return home for a few years at least.)

Any help on any of this, or at the very least, where I can obtain some legal "fa piao" would be appreciated.

The way ur boss worked is outlawed. Obviously, education is not inlcuded in the businesses his company is licensed.
Those fa piao u mentioned is for catering industry. But for some tax reason, they use the fa piao to work out the school's balance sheet. Ur boss must be a middle man who takes profit by providing the fa piao.
And the school is also to blame. The practice, if noticed by tax authority, will face severe fine.
I suggest u not do it directly. Just ask for a raise.