BigQuery StandardSQL: Last 7 Days using _TABLE_SUFFIX

Elliott's answer is correct, but if you want to get the most performance out of BigQuery for such kind of query, instead of converting _TABLESUFFIX to DATE, you should convert CURRENT_DATE expressions to strings:

WHERE
  _TABLE_SUFFIX BETWEEN 
  FORMAT_DATE("%Y%m%d", DATE_SUB(CURRENT_DATE(), INTERVAL 7 DAY)) AND
  FORMAT_DATE("%Y%m%d", DATE_SUB(CURRENT_DATE(), INTERVAL 1 DAY))

You were almost there with your last attempt. You need to convert your string to a date in order to use it in the comparison:

WHERE
  PARSE_DATE('%Y%m%d', _TABLE_SUFFIX) BETWEEN 
  DATE_SUB(CURRENT_DATE(), INTERVAL 7 DAY) AND
  DATE_SUB(CURRENT_DATE(), INTERVAL 1 DAY)