Is there a way to make SQL Server's Database Tuning Advisor use more than one core?
I expect the analysis is carried out serially, one query at a time, as doing so in parallel would be unreliable.

DTA produces recommendations by creating hypothetical indexes and evaluating the impact on a query's execution plan. If analysis were carried out on multiple queries at a time, the index created for one query could influence the analysis of another.

A hypothetical index is created using the undocumented command CREATE INDEX WITH STATISTICS_ONLY and as the name implies this creates just the statistics for the index, without building the physical structure.