A change in the ratio of full-time to part-time employment makes no difference in a country's unemployment rate. However, such a change would certainly affect how we should assess the strength of the economy.
The unemployment rate in a country does not distinguish between full-time and part-time workers. Anyone who has a paid job of any sort is employed. Therefore, an economy where half of the labor force works part-time could have the same unemployment rate as if none of the labor force worked part-time (so long as the same percent of all workers in the labor force had a job).
Of course, the economy with fewer part-time jobs would most likely be the healthier economy. Therefore, when there is a change in this ratio, we should assess the unemployment rate differently. We should note that a change to a higher part-time to full-time ratio would typically mean that an economy is struggling, even if the actual unmployment rate remains unchanged.