It's difficult to argue that there are only two major parts of the cell, especially since so many cells are structured differently, and cannot function without a significant number of components.
The genetic material is undoubtedly the most important, but on its own it isn't especially significant as a "part" of the cell because it's so small and doesn't compose much of the cell's volume. Its particular arrangement also depends on whether the cell is eukaryotic or prokaryotic. We also can't establish it as a significant component because it isn't complimentary or cooperative with any single structure that would make a good "two most important" pairing, unless you wanted to consider the entire metabolism as a "part".
Think the answer this question is looking for is the cell body and the membrane. The body includes the cytoplasm and is essentially the site of the majority of the cells metabolic processes. The membrane is what makes the cell a cell; it insulates the body from its surrounding environment and creates the conditions necessary for many of the metabolic functions to take place.
If they are just asking for two, (which is hugely understating the number of components) then I would go with Nucleus and Cytoplasm.
The Nucleus and Mitochondria are the two major parts of the cell. The nucleus contains the DNA (genetic material) of the cells and this is very important. It is also where ribosomes are created. The Mitochondria creates energy for the cell through the food that is taken in. Energy and DNA are two important things to the cell. And therefore, the Nucleus and Mitochondria are major parts of a cell.
Nucleus and Mitochondria.
It's difficult to say which two parts of a cell are the most important considering each part is necessary for a cell to function. However, I'd say the two most important parts of any cell are the nucleus and the mitochondria. The nucleus is known as the "control center" of the cell and contains the DNA in the cell. The mitochondria is the "powerhouse" of the cell. It turns food into energy for the cell. Both plant and animals cells have these components.
Depends which cell type they are asking for. A plant cell differs from an animal cell such that it has chloroplasts and a cell wall. If asked about the two major parts of a cell, discussing about the nucleus and the cell membrane is a good idea. However, it must be noted that there are many different parts of a cell and that stating only two "major" parts would be undermining the rest of them. Each part of a cell plays a major role in its shape and the way the cell functions.