My parents let me go to Japan alone
My parents let me go [after me asking] to Japan alone
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くれる basically means "to give", and もらう basically means "to receive". Let's think about this with simpler examples:
In the former, the subject is 両親, and in the latter, the subject is 私. Both has に, but these two に play different roles. When you use もらう, the に-marked argument marks the giver. This is a basic rule of how もらう works.
Now, if we replace 本を with 一人で日本に行かせて, we get these:
These are the sentences you are asking about. In the latter, you can see the に-marked argument still marks the giver (of favor).