Is there any computer parts that can be trashed safely (for the environment)?

It is best to drop them off at a local recycler, or donate them to Goodwill or other charity, they will test them, if they are good they will be sold, if not they will recycle them for you.

Our town has a recycling center specifically for electronics. You'd need to ask place like that. Also, different recycling centers may give you different answers.

To cite petersohn comment:

Most computer parts has electric circuits, which usually contain lead. This applies to all electronic equipment, not just computer parts.

Since lead is poison, we can safely assume that we can't trash 99.9% of computer parts.

As others pointed out, it's wiser to donate them (if in working order) to charities or any people who will use them.

Most cities (at least in North America) have recycling centers where you can safely dispose of these "dangerous" things.