PAR files use ReedSolomon error correction which IMO might as well be magic.

Galois Fields are really awesome (and are related to CRC codes). The level of effort to learn is quite high. NASAs guide to ReedSolomon was amazing though

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XOR codes are easier and sloppier. But are actually what's used today despite being imperfect.

Let's say you have A, B, C... Z as your data.

Parity#1 could be XOR(A, B, Z). If B were missing, Parity#1 XOR A XOR Z can back-calculate B.

Parity#2 can be a random set of all previous entries (including Parity#1). Etc. etc. etc.

Keep adding XOR parity codes until your probability of reconstruction is high enough to please you.

I believe this concept is called a Fountain Code.