A cell contains the same DNA as its parent cell. This is a result of which process?

The answer is Mitosis.

Mitosis is a cell division process by which somatic cells reproduce.

In this process there is one division per cycle (with these steps:prophase, metaphase, anaphase e telophase, cytokinesis), one cell produces 2 new daughter cells and their genetic information is the same as the original cell (the number of chromosomes is also the same)

The process in which a parent cells splits into two genetically identical "daughter" cells is called mitosis. This process should not be confused with meiosis.