The immediate problem is that the command substations are expanded inside the double quotes when the alias is defined. Using single quotes will fix that by deferring the command substitution until you actually use the alias.
alias gup='git pull origin $(git rev-parse --abbrev-ref HEAD)'
However, you can save yourself the trouble of trying to get the quoting just right by using a shell function instead.
gup () {
git pull origin $(git rev-parse --abbrev-ref HEAD)
}