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If the fixed point is the intersection of the celestial equator and the hour circle that intersects the body's position on the celestial sphere, it is declination.
The only line that runs parallel through the equator is THE EQUATOR. [The Tropic of Cancer and the Tropic of Capricorn are parallel to the equator, but are north and south of it (respectively) at 23.5 degrees. So they do not run through the equator.] The lines of longitude all run through the equator, but they are not parallel to each other since they all meet up at both the North and South Poles.
The locus of all points of zero latitude ... sometimes also known as the "Equator" ... is the longest parallel of constant latitude. It's length is the equatorial circumference of the earth ... roughly 24,900 miles (40,073 km) .
A single line cannot be perpendicular or parallel. These are attributes of two (or more) lines.
Lines of latitude.