Welcome to the forum.
If you're using the most up-to-date version of Excel, the easiest solution in MINIFS. There are other solutions for old Excel (prior to 2013 version).
I included error-checking to account for Part Numbers you might look up that are not in the Data in Columns A and B, in which case the formulas report blanks ("").
| A | B | C | D | E | F |
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Part | Date | Part | Earliest (MINIFS) | Earliest (AGGREGATE) | | |
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<tbody>
[TD="align: center"]1[/TD]
[TD="align: right"][/TD]
[TD="align: center"]2[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]7/18/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]1[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/18/2016[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/18/2016[/TD]
[TD="align: center"]3[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]7/19/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]2[/TD]
[TD="bgcolor: #E2EFDA"][/TD]
[TD="bgcolor: #E2EFDA"][/TD]
[TD="align: center"]4[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]7/20/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]3[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/20/2016[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/20/2016[/TD]
[TD="align: center"]5[/TD]
[TD="align: right"]5[/TD]
[TD="align: right"]7/21/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]4[/TD]
[TD="bgcolor: #E2EFDA"][/TD]
[TD="bgcolor: #E2EFDA"][/TD]
[TD="align: center"]6[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"]7/22/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]5[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/21/2016[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/21/2016[/TD]
[TD="align: center"]7[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"]7/23/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]6[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/22/2016[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/22/2016[/TD]
[TD="align: center"]8[/TD]
[TD="align: right"]8[/TD]
[TD="align: right"]7/24/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]7[/TD]
[TD="bgcolor: #E2EFDA"][/TD]
[TD="bgcolor: #E2EFDA"][/TD]
[TD="align: center"]9[/TD]
[TD="align: right"]8[/TD]
[TD="align: right"]7/25/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]8[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/24/2016[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/24/2016[/TD]
[TD="align: center"]10[/TD]
[TD="align: right"]8[/TD]
[TD="align: right"]7/26/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"]9[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/27/2016[/TD]
[TD="bgcolor: #E2EFDA, align: right"]7/27/2016[/TD]
[TD="align: center"]11[/TD]
[TD="align: right"]9[/TD]
[TD="align: right"]7/27/2016[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
</tbody>