Generally no, because you'd only be competing against yourself. If you are an attacker, you want to put maximum resources on building one best chain.
For Peercoin the POS mining (called minting) only happens once a second, lasting perhaps less than 100 microseconds (I guess). 99.99% of the time the computer is not minting. Maintain a 10-block altchain (in addition to the main chain) takes less than 100KB RAM (my guess). It doesn't cost that much. But the cost is not absolute zero, either.