I'm not sure what kind of palm you have
but it's not a parlor palm. But whatever Palm it is, it's gorgeous!
For the spidermites, try a few drops of a mild dish soap and water solution in a spray bottle either spray the fronds outdoors if it's not too cold, or in the tub (I would cover the soil) with something like plastic Walmart bags.
Or, you can try using about an inch of rubbing alcohol, the rest water in a spray bottle. Spray every leaf including underneath until it looks like it's raining from the palm. If you miss any of those nasty mites, they will come back!
You might have to do this treatment once a week or so for a few weeks.