Is there a symbol for antiparallel?

I belive that when we talk about vectors you can use $a \uparrow\uparrow b$ for parallel vectors and $a \uparrow\downarrow b$ for antiparallel.

I think you should consider using a symbol that shows your meaning of "parallelism". Could try these two couple: $v \mathrel{{{\upharpoonleft}{\downharpoonright}}} w$ or $v\mathrel{\uparrow\downarrow}w$ as opposite to positively-parallel: $v\mathrel{{{\upharpoonleft}{\upharpoonright}}}w$ or $v\mathrel{\uparrow\uparrow}w$ Hope it helps

There isn't such a symbol (as far as I know) because if $v\|w$ then $-v\|w$ as well. Indeed each vector is parallel to its own opposite, as you take only the line on which it lies, when you speak of direction. (To describe that fact you however can write $v\in\mathbb{R}^{-}w$ in an opposite way of "positive" parallelism:$v\in\mathbb{R}^+w$)