Finding parameters of image filter using classified pairs
Your parameter $\alpha$ has fairly low dimension.  Therefore, I recommend that you apply optimization methods directly to try to find the best $\alpha$ (without trying to use convolutional neural networks and regression for this purpose).

Define a distance measure on images, $\|I-J\|$, to represent how dissimilar images $I,J$ are.  You might use the squared $L_2$ norm for this, for instance.

Now, the loss for a particular parameter choice $\alpha$ is

$$L(\alpha) = \sum_{k=1}^N \|F_\alpha(I_k)-J_k\|.$$

We can now formulate your problem as follows: given a training set of images $(I_k,J_k)$, find the parameter $\alpha$ that minimizes the loss $L(\alpha)$.

A reasonable approach is to use some optimization procedure to solve this problem.  You might use stochastic gradient descent, for instance.  Because there might be multiple local minima, I would suggest that you start multiple instances of gradient descent from different starting points: use a grid search over the starting point.  Since your $\alpha$ is only three-dimensional, it's not difficult to do a grid search over this three-dimensional space and then start gradient descent from each point in the grid.  Stochastic gradient descent will allow you to deal with fairly large values of $N$.

This does require you to be able to compute gradients for $L(\alpha)$.  Depending on the filter $\alpha$, it might be possible to symbolically calculate the gradients (perhaps with the help of a framework for this, such as Tensorflow); if that's too hard, you can use black-box methods to estimate the gradient by evaluating $L(\cdot)$ at multiple points.

If $L_2$ distance doesn't capture similarity in your domain, you could consider other distance measures as well.

I expect this is likely to be a more promising approach than what you sketched in the question, using convolutional networks and a regression model.  (For one thing, there's no reason to expect the mapping from "features of $I_k$" to "features of $J_k$" to be linear, so there's no reason to expect linear regression to be effective here.)