During Regular Weekly Customisation Spotlight rounds, you may enter using Pets from your side accounts. The only prizes for the Regular Weekly Contests are the trophies and avatar, so this is acceptable to participate in on your side accounts.
However, Themed bi-weekly Customisation Spotlight rounds award special NP and NC prizes, so side accounts are not eligible to enter into the Themed Contests. If the current round is themed, you will need to enter from your main account only.
So to summarise: You cannot enter the Themed Contests on your side accounts, but you can continue to enter your Neopets from your side accounts into the Regular Contests.