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Aren't later equity rounds usually senior to earlier rounds (the "last money in, first money out" principle) anyway?

No, not unless the later equity round is in a different type of security (preferred equity, convertible debt, etc.).

Otherwise, all holders of common equity are considered to be on equal footing (pari passu)

Don't investors normally get preferred shares?

Not unless you agree to sell preferred shares. All up for negotiation.