The answer to your question is No. 

Here is the reasoning.

EIP #1 for $500 was allowed under Code Sec. 6428, as added by the CARES Act, and EIP #2 for $600 under Code Sec. 6428A, as added by COVIDTRA. Both code sections define a qualifying child as defined under the Child Tax Credit rules of Code Sec. 24(c), which says “The term “qualifying child” means a qualifying child of the taxpayer (as defined in section 152(c)) who has not attained age 17.”

The rules of Code Sec. 151 and Code Sec. 152 remain effective for 2020, even though the deduction for dependency exemptions does not apply.

 To claim EIP #1 and/or EIP #2, Parent 1 would have to claim the child as dependent. Since Parent 2 has already claimed the child as dependent on their 2020 tax return, Parent 1 is barred from doing so on their 2020 tax return.

In other words, the EIPs are tied to the dependent – whoever claims the qualifying  child as dependent gets the EIPs.