Is "We have the power, so can you" (from She-Ra) grammatical?

The lyrics of the closing theme of the She-Ra cartoon include the line "We have the power, so can you". (Listen to it on YouTube.)

I'll grant them their poetic license, but it doesn't sound right to me and I can't quite put my finger on why.

Would you say it's ungrammatical? Why?

The example you quoted has weak parallelism.

It also reminded me of Stephen Colbert's book, I am America (and so can you!).

prashnrao's answer led me to a comment by eds in Please explain "I Am America (And So Can You!)":

The part where it becomes confusing, and funny, is that the verb is not an action verb, but "am." So "...so can you" means "You can be America too" in this case.

That may be it – I suppose "having the power" is not quite an action that you can perform. Just substituting a more actiony verb like "we use the power, so can you" sounds much better to my ears.

It would be satisfying if there was some means of determining just what verbs are acceptable, other than trying them in a sentence, but I can't think of any, and I suppose there might not be any.