Yes, you can include your share of the employment taxes as qualified expenses.
According to Household Services in the 2021 Instructions for Form 2441,
You can also include your share of the employment taxes paid on wages for qualifying child and dependent care services.
Also, see the 2021 Instructions for Schedule H for information on your share of employment taxes.
[Edited 1/26/2021 | 5:29 am PST]