I've investigated and found two articles with no answer for this question:
It's been a while, so maybe something could have changed, so raising this question again. I am wondering if there's a way to query Salesforce to get the date of a sandbox refresh. The only thing I've found that comes close is through using this query:
Select createddate from SetupAuditTrail order by createddate asc limit 1
I am wondering if this is still the only viable solution.