Any subquery in the select-list has to be a scalar subquery, that is it must have one column and one row.
But in this case you don't need a subquery at all:
SELECT
Office,
COUNT(Office) As Tot_Part,
SUM(completion_status IN ('Started', 'Complete')) AS Total_Resp
FROM trespondent
WHERE completion_status <> 'New'
GROUP BY Office
ORDER BY Office
The trick is that SUM() of values that are 1 or 0 is the same as COUNT() of the rows where the value is 1. And MySQL treats boolean expressions as 1 or 0.