1. A traveling road surface detection apparatus that detects a traveling road surface on the basis of a parallax image around a vehicle, the parallax image being acquired by an in-vehicle camera, the traveling road surface detection apparatus comprising: an electronic control unit configured to i) determine using parallax information of the parallax image whether a unit area is a road surface area or a non-road surface area for each unit area of the parallax image, each unit area including one or multiple pixels, ii) associate each unit area with a grid in a map on the basis of parallax information, and coordinate position of the unit area, an eye point of the map being set so as to be higher than an eye point of the parallax image, the map being obtained by partitioning an area around the vehicle in a grid, and iii) detect road surface grids and each grid located between the road surface grids on the map as the traveling road surface on the map, the road surface grids being the grids corresponding to the unit areas determined as the road surface areas.