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I understand your question to be, "Is it true that parallel lines are everywhere equidistant and never intersect?"

In what follows, I assume we're talking about a two-dimensional plane.

By definition, two lines that are parallel (in the same plane) never intersect. In Euclidean (AKA Parabolic or simply E) Geometry, and also in Hyperbolic (AKA simply L) Geometry, parallel lines exist. In Elliptical (AKA R) Geometry, all lines eventually intersect so parallel lines do not exist.

Now, are two parallel lines (in the same plane) everywhere equidistant? If so, that means that it is possible, at any point on one of the lines, to construct a perpendicular that will meet the other line in a perpendicular, and that the length of the segments constructed will be always the same.

In Euclidean Geometry, two parallel lines (in a plane) are indeed everywhere equidistant. To prove it requires the converse of the Alternate Interior Angles theorem (AIA), which says that if two parallel lines are cut by a transversal, the alternate interior angles will be congruent. Note that this is the CONVERSE of AIA, not AIA. Some people get this mixed up.

In Hyperbolic Geometry, two lines can be parallel, but be further apart some places than others. I know that sounds rather odd, if you're not used to it. Here's an image that might help: imagine that your plane is a thin sheet of rubber, and for some reason is being stretched. The further you go from your starting point, the more it stretches, and it's always stretching away from you. This means that your parallel lines will keep getting further and further apart.

Q: Parallel lines are equidistant and will never meet?

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The answer would be parallel lines these lines never meet or cross each other.