Relative vs. Parallel Modes
Any two modes are related to each other in one of two ways, depending on what you hold fixed. Keep the notes the same and move the tonic, and you get relative modes. Keep the tonic the same and move the notes, and you get parallel modes.
Relative: C Ionian and F Lydian
C Ionian and F Lydian are built from the exact same seven notes - the white keys on a piano - just started from a different one: F is the fourth note of that same C Ionian scale, and starting those seven white keys there instead gives Lydian. Because the note collection is identical, the pattern of whole and half steps between neighboring notes is identical too, at the same physical points around the wheel. Only the flag marking the tonic moves.
That's the test for whether two modes are relative: if every colored arc lines up in the same spot on the wheel and only the tonic flag differs, the notes never changed - you only rotated which one counts as home.
Parallel: C Ionian and C Lydian
C Ionian and C Lydian keep the same tonic, but they are not the same seven notes. They differ on exactly one degree - the fourth, natural in Ionian and sharp in Lydian - which is the whole reason Lydian has its floating, unresolved quality.
The wheel makes the difference positional rather than verbal: the fourth-degree wedge sits one wedge further clockwise in Lydian, and the spoke drawn from the center recolors to the interval that degree now forms with the tonic. Everything else on the two wheels is identical.
Unlike relative modes, which always share every note by definition, parallel modes can differ by as little as one note or by several, depending on which two modes you pick. That “sharp 4” label is a scale degree - every mode gets described this way, as Ionian's seven degrees with some subset raised or lowered, which is exactly what a parallel comparison measures. For a closer look at this exact pair, see Major vs. Lydian, or Comparisons for others like it.
Notice both examples above reach for the same two modes, Ionian and Lydian - only the tonic assigned to Lydian changes. Give it F, and it's relative to C Ionian: same notes, different home. Give it C instead, and it's parallel: same home, different notes. Relative and parallel aren't opposites so much as two different questions you can ask about the same pair of modes - for the fuller picture, including how the modes are numbered and why relative modes always share the same chords, see How the Greek Modes Relate to Each Other.