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In my opinion:

walk beside me means walk should-to-shoulder with me.

The following link has a picture, which will further let you know what I think about walk beside me:

In that picture a girl and a boy are standing beside to each other.

Few days ago, I and my friend were cycling. My friend little ahead of me. Suddenly, he asked to me ''Jack, come forward and cycle parallel to me''. By parallel he meant to say that I cycled shoulder-to-shoulder with him, so that we could also chat while cycling. Is the use of word parallel right here? Shouldn't he had used beside in stead of parallel in cycle parallel to me? Don't you think using word parallel look mathematical?

Comments  
should-to-shoulder
means closer IMO than in that pic, those shoulders should be practically touching each other, like in the army

cycle parallel to me
means
cycle on a path which is parallel to mine, but not necessarily on the same level, could be ahead, could be behind a bit
I think the most natural thing to say is "Come and ride next to me so we can talk."
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Jackson6612

In my opinion:

walk beside me means walk should-to-shoulder with me.

The following link has a picture, which will further let you know what I think about walk beside me:

In that picture a girl and a boy are standing beside to each other.

Few days ago, I and my friend were cycling. My friend little ahead of me. Suddenly, he asked to me ''Jack, come forward and cycle parallel to me''. By parallel he meant to say that I cycled shoulder-to-shoulder with him, so that we could also chat while cycling. Is the use of word parallel right here? Shouldn't he had used beside in stead of parallel in cycle parallel to me? Don't you think using word parallel look mathematical?

Hi,

In my town phrases like cycle parallel to me and drive parallel to me are very common. In these phrases parallel means to cycle/drive on the left or right side of someone. Is parallel a suitable word to use? Could beside or alongside be used in stead of parallel in those phrases?

Isn't "parallel" extremely broad a term compared to those others? A dude could be riding his bike ten miles to your left or right and if he was going exactly in the same direction as you, he'd still be "cycling parallel" to you, no?
OcelotIsn't "parallel" extremely broad a term compared to those others? A dude could be riding his bike ten miles to your left or right and if he was going exactly in the same direction as you, he'd still be "cycling parallel" to you, no?
Yes, you are correct.
Students: We have free audio pronunciation exercises.
Yes, parallel isn't related to distance or relative distance (ahead, behind).
Jackson6612Hi,

In my town phrases like cycle parallel to me and drive parallel to me are very common. In these phrases parallel means to cycle/drive on the left or right side of someone. Is parallel a suitable word to use? Could beside or alongside be used in stead of parallel in those phrases?

Well, like I already say, "Come up and ride next to me" -- or "beside me" -- sound natural. If where you live, people do say "parallel," then when you are at home, you should use it. However, if you come to the U.S. and want someone to come up and ride next to you and you say "Come ride parallel to me" the person would probably think you had an obsession with math and would probably take a moment to understand what you meant.