A couple of issues to address here.
First, parallel LM386s is not the best idea. If you parallel before the output cap then the imbalances between the chips will drive the bias points and one or both chips will have unbalanced action. Think HEAVY 3rd harmonic distortion. Regardless of before or after the output cap, the feedback between stages will be unbalanced and you'll likely get oscillation. Even if it does comes out stable, the distortion will be uncorrected by the feed back and very high.
Second, the N package (DIP) is rated for 1.25W with a 107ºC/W JA thermal resistance. You'll have to do the calculations to see how far you can drive dissipation in your assumed operating temperature. A DIP heat sink may be a good solution. However, regardless of this, the 386 is only capable of at best 350mW into a 4Ω load. It will go over a watt into a 16Ω load which is why the 386 makes such a fine headphone amp (according to some, not to others. Read this for my experience, 4th paragraph down.
). But attempting to get a watt into 4Ω you'd be better to use the 386 to drive a Darlington pair biased for class A operation. Or you could try a different chip more suited to higher output.
The bottom line is that the LM386 was always intended as a small, low power solution for portable radios, small consumer goods, and the like. Personally, if would suggest that if you're going to go to the effort to build something that looks half way decent, you should invest in a chip more up to the task at hand. IMHO.
I've used the 386 for a long time for all kinds of projects just because it's a simple one chip, no thought solution to providing an audio output on a circuit. But as a dedicated amp, I have always found it somewhat lacking.