Economists typically say that there are three distinct types of unemployment. These are structural, frictional, and cyclical. (Some add a fourth type, seasonal.) Of these, only one tends to go up due to recessions. Only cyclical unemployment rises because of recessions.
Structural unemployment occurs when people lose their jobs because their jobs are no longer needed in an economy. For example, when people started listening to music on CDs, the jobs of people who made vinyl records became relatively obsolete. This sort of thing is not caused by recessions. It is caused by changes in technology and changes in tastes.
Frictional unemployment occurs when people are just entering the workforce or when they quit a job or get fired (as opposed to laid off). Recessions do not cause people to enter the workforce in larger numbers. They do not cause people to quit their jobs or to get fired. Therefore, recessions do not cause frictional unemployment.
Cyclical unemployment can be defined as unemployment that comes about because there is not enough economic activity going on. In other words, cyclical unemployment happens when people in a country are just not buying as much as they usually do. Recessions do cause cyclical unemployment because they cause people to have less money and to buy fewer things. Therefore, cyclical unemployment is the type of unemployment that is caused by recessions.