Remember that "component" can be an adjective or a noun. As a noun, it is sometimes interchangeable with the word "part", especially when talking about mechanical components, such as parts of an engine, so I can see why you might think it is a tautology to say both words. But as an adjective, "component" describes things that go together to make up a whole, whereas a "part" of something does not necessarily fit that description alone and may need the adjective to describe it.
For example, consider a cake. If you take a knife and divide a cake into slices, you could say that each slice is part of the cake, but you would not say that those slices are the components of a cake - the components of a cake are flour, sugar, eggs and milk.
In your example of breaking down a sentence, it does not seem unusual to me to use the term "component parts" because it better describes functioning sections of a sentence than either word alone. What is "part" of a sentence? It could be one word, or two words that in isolation carry no meaning. What are the "components" of a sentence? You probably heard it said that every sentence must have a subject and a verb, but it can have other things too. There are rules of sentence construction but no exact blueprint like there is for an engine. So, when you are looking at a sentence that has already been constructed and are being asked to break that down into component parts it is asking you to decide where to break it up into sections (parts) that could be analysed individually, but they are component parts because they will go back together to form the original sentence.