Sumif for multiple criteria in one row and two columns

Hi, in the attachment, I'm wondering how to take values in the top table, sum them up only if they're in the "X" rows, and then return the sum of all the numbers by month in the bottom table.

Example:
For part # B, I want to output the sum of 30+31+30+31 = 122 and put the result in the column for February for Part B.

It seems like a simple concept, and I've tried vlookups, sumifs, and sumproducts with criteria, but I can't figure out how to make it work.

Thank you guys so much!