ImageDataGenerator for multi task output in Keras using flow_from_directory
Have you tried creating a custom Data Generator for your use case? The structure for it can be found below. You will need the __data_generation method such that you return y as an array with 2 elements, with the first element being your labels for task 1 and the second element for task 2.

import numpy as np
import keras

class DataGenerator(keras.utils.Sequence):
    'Generates data for Keras'
    def __init__(self, list_IDs, labels, batch_size=32, dim=(32,32,32), n_channels=1,
                 n_classes=10, shuffle=True):
        'Initialization'
        self.dim = dim
        self.batch_size = batch_size
        self.labels = labels
        self.list_IDs = list_IDs
        self.n_channels = n_channels
        self.n_classes = n_classes
        self.shuffle = shuffle
        self.on_epoch_end()

    def __len__(self):
        'Denotes the number of batches per epoch'
        return int(np.floor(len(self.list_IDs) / self.batch_size))

    def __getitem__(self, index):
        'Generate one batch of data'
        # Generate indexes of the batch
        indexes = self.indexes[index*self.batch_size:(index+1)*self.batch_size]

        # Find list of IDs
        list_IDs_temp = [self.list_IDs[k] for k in indexes]

        # Generate data
        X, y = self.__data_generation(list_IDs_temp)

        return X, y

    def on_epoch_end(self):
        'Updates indexes after each epoch'
        self.indexes = np.arange(len(self.list_IDs))
        if self.shuffle == True:
            np.random.shuffle(self.indexes)

    def __data_generation(self, list_IDs_temp):
        'Generates data containing batch_size samples' # X : (n_samples, *dim, n_channels)
        # Initialization
        X = np.empty((self.batch_size, *self.dim, self.n_channels))
        y = np.empty((self.batch_size), dtype=int)

        # Generate data
        for i, ID in enumerate(list_IDs_temp):
            # Store sample
            X[i,] = np.load('data/' + ID + '.npy')

            # Store class
            y[i] = self.labels[ID]

        return X, keras.utils.to_categorical(y, num_classes=self.n_classes)


You can then use your custom generator with the Model object you have created as shown below: 
training_generator = DataGenerator(partition['train'], labels, **params)
validation_generator = DataGenerator(partition['validation'], labels, **params)

# Design model
model = Sequential()
[...] # Architecture
model.compile()

# Train model on dataset
model.fit_generator(generator=training_generator,
                    validation_data=validation_generator,
                    use_multiprocessing=True,
                    workers=6)


The codes have been taken from https://stanford.edu/~shervine/blog/keras-how-to-generate-data-on-the-fly