are lines of latitude horizontal or vertical?

They may be thought of as 'horizontal'.
Lines of longitude (meridians) may be thought of as 'vertical'.

As we liver on a Sphere , it is irrelevant to think of these lines as 'horizontal and vertical', It is just a matter of clarity in cartography ( map making).

Lines of latitude cross lines of longitude at 90 degrees at all points on the Earth's surface.

Latitude is parallel to the equator so on an atlas parallels of latitude are horizontal.

Latitudes,also known as parallels are Horizontal lines........They never intersect each other...