Currently, a subjective method is used to diagnose cough sounds, particularly wet and dry coughs, which can lead to incorrect diagnoses. In this study, novel emergent features were extracted using spectrogram methods and a parallel-stream one-dimensional (1D) deep convolutional neural network (DCNN) to classify cough sounds. The data of this study were obtained from two datasets. We employed the Mel spectrogram, chromagram constant-Q transform, Mel-frequency cepstral coefficient, constant-Q cepstral coefficient, and linear predictive code coefficient to conduct features analysis. The maximum, mean, variance, and standard deviation values of the original spectrogram as well as the maximum first and second derivatives of this spectrogram were extracted and fused to create a single-feature vector. We adopted two types of features: single features and combined features. Each design was restructured according to the magnitude of features with high discrimination power. A parallel-stream 1D-DCNN was developed for classifying cough sounds accurately. We compared the results obtained using the aforementioned network with those obtained using a single-stream 1D-DCNN. We found that the parallel-stream network outperformed the single-stream network for some feature sets. The developed network achieved F1 scores of 98.61% and 82.96% for the first and second datasets, respectively. The concatenation of layers at the flattening level resulted in an F1 score of 99.30% in dataset one. Moreover, layer merging strategies exhibited a better performance at the second convolutional layer level than at the flattening layer level in many cases.