Speed up/cache directory size computation in health check.

The output directories are growing to several gigabytes in size, which
is causing the health check to slow down.  Add back some parallelism
and cache results more aggressively in order to make the health check
fast again.

Bug: 71260
Change-Id: I2fddb23a7fd2f05dcf96455793475e7ae5e7b4b2