Ignoring the comment about parents "letting" their child claim their own exemption ... 😂
Yes, EIC is allowed in SOME cases when the parent does not claim the child as a dependent. The support test does not apply to EIC. So if the child would qualify as a dependent (including living with parent over 1/2 of the year) EXCEPT for the fact the child paid for 1/2 of their own support, EIC is still allowed.
Likewise, it should apply if the parent chooses not to claim the child, but is eligible to (I'm not sure of a situation where that would apply, but it seems hypothetically possible).