Cómo usar la condición BETWEEN usando la consulta principal y las subconsultas

Tienes que agregar WHERE cláusula a cada subconsulta de la siguiente manera:

  SELECT COUNT(T.tripId) as Escort_Count,
  (
      SELECT COUNT(*) FROM 
      (
          SELECT a.allocationId
          FROM escort_allocation a 
          INNER JOIN trip_details b ON a.allocationId = b.allocationId 
          INNER JOIN cab_allocation c ON a.allocationId = c.allocationId 
          WHERE c.allocationType = 'Adhoc Trip' AND tripDate BETWEEN '20180315' AND '20180320'
          GROUP BY a.allocationId
      ) AS Ad

  ) AS Adhoc_Trip_Count,
  (SELECT COUNT(id) FROM trip_details WHERE tripDate BETWEEN '20180315' AND '20180320') as Total_Count
  FROM 
  ( 
      SELECT a.tripId FROM 
      trip_details a 
      INNER JOIN 
      escort_allocation b 
      ON a.allocationId = b.allocationId 
      WHERE tripDate BETWEEN '20180315' AND '20180320'
      GROUP BY a.allocationId 
  ) AS T