Classification of degraded images is very important in practice because images are usually degraded by compression, noise, blurring, etc. Nevertheless, most of the research in image classification only focuses on clean images without any degradation. Some papers have already proposed deep convolutional neural networks composed of an image restoration network and a classification network to classify degraded images. This paper proposes an alternative approach in which we use a degraded image and an additional degradation parameter for classification. The proposed classification network has two inputs which are the degraded image and the degradation parameter. The estimation network of degradation parameters is also incorporated if degradation parameters of degraded images are unknown. The experimental results showed that the proposed method outperforms a straightforward approach where the classification network is trained with degraded images only.