If you spray your tent with Permethrin to protect yourself from Lyme-disease-carrying ticks and blood-sucking mosquitos, you should spray the inner tent or insect netting and not the rain fly, which is waterproof and will prevent the Permethrin from soaking in. Permethrin is also quickly broken down by ultra-violet light, so it will break down quickly with sun exposure.
If you decide to treat the inner non-waterproof part of your tent, by spraying or soaking, you want to avoid using Permethrin or Permethrin Concentrate containing petroleum distillates and use water-based Permethrin instead. The petroleum distillates may degrade or ruin any synthetic waterproof coatings on your tent. If in doubt, call the tent manufacturer and ask for their advice. You should also avoid using Permethrin with petroleum distillates on clothing since it can cause skin reactions and can leave an unpleasant odor that lasts a long time. Unfortunately, repeated washing to get rid of the odor will also get rid of the Permethrin.
The only water-based Permethrin or Permethrin Concentrates I’m aware of is Sawyer Permethrin or Martin’s Permethrin 10% Concentrate, although some different formulations of Martin’s do include petroleum distillates, so you should avoid those. Generally speaking, the Permethrin sold in feed stores has petroleum distillates and is not intended for use on human clothing or tents, but for cattle or horses who are more tolerant of it. When in doubt, read the label or search online for the safety data sheet, which lists the ingredients.
Even before you spray your tent, you should spray or soak your clothes with Permethrin (which will last for 6 wash/dry cycles) or send them to Insect Shield for treatment (which will last for 70 wash/dry cycles). Ticks latch onto your shoes and clothes when you walk through vegetation or forest duff and this is the most likely scenario for encountering one.