Which parent can claim the Education Expense Tax credit for child in K-12 grade?

First of all there are not any education credits for grades K-12 on a federal tax return.   And you cannot get education credits when you file married filing separately even  if the student is in college.

EDUCATION EXPENSES

Education expenses for grades K-12 are not deductible on your Federal return, whether for private, public, or home schooling, nor are the expenses for tutoring, after school lessons or after school activities, such as dance lessons, sports, etc.  Some states allow deductions/credits for K-12 education and/or home schooling expenses; if your state has these deductions available, you will be prompted to enter them when you prepare your state return.  (As far as I know, the states that offer any sort of K-12 deductions/credits are Arizona,  Illinois, Indiana, Iowa,Louisiana, Minnesota, and Wisconsin)

And....why are you filing separate returns?   That is usually the worst way to file.

If you were legally married at the end of 2020 your filing choices are married filing jointly or married filing separately.

Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will get the married filing jointly standard deduction of $24,800 (+$1300 for each spouse 65 or older)  You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit. 

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return. Some of the special rules for filing separately include: you cannot get earned income credit, education credits, adoption credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. Your limit for SALT (state and local taxes and sales tax) will be only $5000 per spouse. In many cases you will not be able to take the child and dependent care credit. The amount you can contribute to a retirement account will be affected. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. ( Community property states:  AZ, CA, ID, LA, NV, NM, TX, WA, WI)

 If  you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice.

https://ttlc.intuit.com/questions/1894449-married-filing-jointly-vs-married-filing-separately

https://ttlc.intuit.com/questions/1901162-married-filing-separately-in-community-property-states

https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separately