Can a permanent resident alien vote?

Legally NO. You must be a CITIZEN of the United States to Vote.
This doesn't mean that it does not happen.

I believe the question and the answer given are somewhat miss-leading. The question was 'Can a permanent resident vote?' And the correct answer is that it depends... As you may know, a permanent resident is not a citizen, which means that he/she does not have all the federal rights, since citizenship is a federal issue. However, the law allows permanent residents to participate in local elections. So, you may not vote for presidents, senators, congress people, or any of those things.

Voting
One of the most important privileges of democracy in the United States of America is the right to participate in choosing elected officials through voting. As a Permanent Resident you can only vote in local and state elections that do not require you to be a US citizen. It is very important that you do not vote in national, state or local elections that require a voter to be a US citizen when you are not a US citizen. There are criminal penalties for voting when you are not a US citizen and it is a requirement for voting. You can be removed (deported) from the US if you vote in elections limited to US citizens.