It's not so much that getting the perm is a problem for health reasons.
It's that your hair isn't going to respond in a predictable manner when you're pregnant.
The hormones of pregnancy change the composition and texture of your hair. So the perm may or may not hold.
What little evidence there is indicates that getting a perm or chemically straightening your hair is no cause for concern. The fumes might be annoying, but the exposure probably isn't significant enough to have any effect on your baby. Some hair-processing solutions may use solvents, and while there is concern about the effects of solvents on a developing baby, that's most likely an issue only for women who work around solvents all day and develop symptoms from the exposure.