Should the tax on repair parts be paid once by me and again by my customers?
Sales tax rules are determined by each individual state, but generally sales tax is only collected from the final end user of the product.
In most states retailers are not required to pay sales tax when they buy from a wholesaler. In the state of California, you can apply for a resale certificate through the Board of Equalization. This certificate exempts you from paying sales tax on items that will be resold to a customer.
In your situation, only your customer should have been charged sales tax. Since you should not have been required to pay the sales tax, you may be able to petition the state for a refund of the sales tax you paid on the repair part.
Of course, this is assuming California state law. You need to verify your sales tax obligations for your specific state.