There's nothing that says that you can't start training to be a CFI from hour 1... With regards to the chute, the Dulles, Richmond, and Fargo FSDO's have all told me that I can conduct spin training without a chute for any pilot whether or not working towards a rating.

AC61-67C backs that up:

b. Because spin entry, spins, and spin recovery are required for a flight instructor certificate or rating, a person receiving instruction from a CFI (or an ATP instructing in accordance with section 61.167) need not wear an approved parachute while instruction is being provided in these maneuvers. This provision applies regardless of the certificate or rating for which the person is receiving training and also if the person is receiving instruction that is not being provided for the purpose of obtaining any additional certificate or rating. The instructor providing the training is also not required to wear an approved parachute while providing this flight training.