For fungus gnats....have you tried using the 'mosquito dunks' they sell in garden centres and hardware stores? They contain BTI which is a bacteria that infects the larva of the fungus gnat. Perfectly safe for humans. Just put a piece of the dunk in your watering can or in a bucket and add water. Let it sit for a day or so and then use the water for your plants. The dunks will last a long time as you can reuse the piece until it's gone.
Another thing you can do is get some insecticidal soap and mix it up in a bucket. Imerse the whole plant, pot and all, into the soapy mix. Remove and let them drain. That works pretty well. Yellow sticky paper also helps to control the problem but won't eliminate it.
As for spider mites....End All from Safer's is very good for those but don't let your pets lick the leaves after spraying. Otherwise, higher humidity discourages them. Try putting your Parlour Palm on a dish with gravel in it and then keeping water in the dish to just below the pot level. Or, misting the foliage daily will also help.

I've become very familiar with spider mite. Last summer I controlled them on my Erythrina by misting twice daily and wiping the leaves daily with a kitchen sponge and a bucket of water.
Phytoseiulus persimilis is a predatory mite which has been highly recommended. Google that for US supply.

Thanks for the help! I'm not having nearly as many fungus gnats buzzing around and I got rid of the spider mites :)