Pumps in a series are installed one after another in the piping. This is usually to achieve the necessary head to pump water from a lower level to a higher level while maintaining pressure standards and waterflow. This is possible because two pumps in series allow their respective heads to be added together to achieve a higher overall head. However, these pumps will pump at a constant (not variable) flow rate.
One disadvantage of series pumps is if one pump goes out, the waterflow is interrupted at that point in the system.
Pumps in parallel are arranged parallel to one another, rather than one after another. This allows each pump to function independently of the others. If one fails, the waterflow will not be interrupted, or at least minimally. Parallel pumping is often used to achieve high waterflow rates. The waterflow rate of each pump can be added to the others to increase this value and match the load. Head values of parallel pumps are not additive, however, as you see in series pumps.