I recently changed my front headlight lamps due to the oxidized condition.I did not disconnect battery.I do not have taillights and the 10 a fuse keeps blowing.Where is the taillight relay located?

Check your connectors & wires for shorts, relays are in the circuit breaker box under hood.

Did the tail lights work before installing the headlights?

yes

There is a relay called a taillight relay . The taillight relay is not located in either fuse box. There must be some differences with the taillight relay with the 1999 and my 2002 Avalon. How can i find out where the taillight relay is on my 2002 Avalon XLS ? I am assuming that i might have fried this relay due to not disconnecting the battery when i changed the headlight assemblies. What do you think ?

I really doubt not disconnecting the battery would have fried it unless you had the lights on while doing it and even then I doubt it.

I think you shorted out a wire to the parking lights by pinching it.

I have front parking lights - brake lights,turn signals. The only circuit that is not functioning is for the taillights.

I hope you are right - The wiring & connections are in pristine condition in the trunk . I do not have wire running along the trunk hinge. On my 02 the tail light wiring is on the right side running through the trunk to the front.