I am sorry this is long... I just need help.

My father is a difficult man to get along with. If even a small mistake is pointed out to him, or a difference of opinion is said, he will get extremely angry. I am not talking violent or emotionally abusive, but nonetheless difficult. He will often storm out of the room and not talk to whoever he is mad at for awhile. As an example while watching a TV show with my mom he pointed out the illogical qualities of the episode. My mom laughingly replied that the series had never had logic to it. My father then proceeded to leave the room, didn't talk to my mom for a couple of days and no longer watches the show they both once enjoyed. It doesn't matter how the critique is phrased or even if it is a critique at all, if my father believes it to be some sort of opposition he will react accordingly. As you may imagine, this makes it hard to point out his anger issues. He does not "believe" in therapy, and if the hint of disapproval about how he handled some situation comes up he becomes very angry. Even a suggestion of how he could do something better makes him furious. I once showed him a new way of peeling an avocado after a cooking class I had and he became angry and refused to finish dinner. In all honesty I was only trying to show him what I learned, not attempt to "prove him wrong." On the other hand, he gets extremely angry if someone does not take his advice on how to complete a project.

Now, I have recently moved away and his issues no longer affect me as much, however being home on holiday made me nervous for my parents marriage. I recognize I am not in charge of my parents, but I still want to help somehow. My mother is obviously tired of not being able to speak her mind, and she doesn't seem to have anyone to turn to. She never talks to me about problems in her marriage as she has a strong belief that children should not have to deal with parents problems, but today she broke down and talked for a short whole about how hard it is for her. I think my father needs help of some kind, but I fear that any attempt on my part to ask him to seek help will only make things worse for my mom. Is there anything I can do? How should I approach the situation?

I am lost and confused and any thoughtful advice given would be greatly appreciated.

First question for me is if this is new or not. If sudden, then somehow your mother has to convince him to have a medical checkup (in some general context, not his anger). He could have HBP or any number of other conditions). Second is his age, which I assume is around 50? He could be winding down in testosterone and sexual ability, and it's upsetting him in ways he doesn't realize. Third is his financial situation and work, which is actually the number one thing men care about, according to many studies. If he hates his job, has lost his job, or is worried about it, then this could easily be a direct cause.
Those are possible reasons, but not excuses. Working on this attitude is a matter for your mother: she has to say be less critical or else, and she has to mean it. They could use couple counseling, in which techniques and exercises are given, not therapy and analysis.
What should you do? Tell your mother that you will spend as much time listening as she can talk to you, and that you will support her, but that you can't be the one to get him to change or get help to change.

This is emotional abuse as your mom is worried about what she can and can not say. I was emotionally abused and this abuse is not recognized as much. The best gift to help your dad is helping your mom. Encourage her to get counseling so she can speak up and tell your dad what she wants in her marriage. I like Joy's advice for your dad but I feel for your mom as I know how much anxiety walking on egg shells can be. It is very draining on a person living in fear that what they say can provoke another outburst followed by withdrawal of communication and love.
There really is not much you can do but encourage your mom to take care of her needs and encourage your dad to seek help by professionals.

You have been living with this behavior a long time, your mom even longer. Maybe she should speak with someone to help her with a coping strategy. Encourage and support her in this.

I would tell your dad you love him, but ask HIM WHY he acts so abusive toward his love ones. He probably doesn't know, or is helpless to stop it, or change without help, if he even wants it. Do you have siblings? Is there no good side of him or is he always a petulant sourpuss? What of other relatives like the rest of his family?