You need to differentiate where you want to apply the tick control. Not on the dogs bodies I assume. There are good medications available for that. Search multiple previous discussions on this forum.
Insecticides for the dog environment can be considered, but permethrin and fipronil are not what you could consider natural, or organic program compatible.
To address your specific question:
Pyrethrins aka pyrethrum is a natural botanical substance, but short lived effectiveness. It biodegrades in 12 to 24 hours. I use it for a contact spray, for knock down of a chewing or sucking plant pest in an organic program, but it's not going to have the residual effectiveness needed for tick control.
Permethrin is a pyrethroid, the synthetic chemical derivative of the botanical pyrethrum. Not natural, but low toxicity for people and pets, and has residual effectiveness. It is used for tick collars and mosquito nets, and plant pest control. For spraying the bedding and dog run areas, do more research on appropriate use, Integrated Pest Management, combinations of insecticides and sanitation measures.
Fipronil is yet another class of chemical insecticide. It's not natural or organic program compatible. This is used in closed system termite control, but should not be sprayed into the open environment, lawn, trees, shrubs. It can be picked up by foraging pollinators and transported back to the hives where it could kill an entire honey bee colony.
Do some more homework on DE, I haven't used it. Search IPM for ticks. There are YouTube videos and articles from many sources available.