You can definitely apply for both at the same time. Canada doesn't really care but the US might not like it if they heard you are applying for Canada PR too but it's not like you are going to tell them? If you get both, it would be hard to keep both though. The US requires that you don't leave the US for more than a year at a time without getting a returning resident permit and if they think you have left the US permanently, they have the right to take your GC away any time. Canada requires you to live in Canada for at least 24 months in any 5 year period. If you had a job where you could work 6 months in each country and filed taxes in both, you might be able to keep both PR's going.
