Translation:My parents replied that I was not at home.
Maybe this is petty, but shouldn't it be "... I am not at home." I'm thinking that someone rings the door bell and asks "Is he at home?" or "Was he at home yesterday?". I think the description of the answer would be different?
This is how the English indirect reported speech works https://dictionary.cambridge.org/grammar/british-grammar/reported-speech/reported-speech-indirect-speech
Wow! Your grasp of both English and Czech is amazing! I stand corrected.
I'm still learning, too, and I actually see a slightly different place to get tripped up... that the literal translation of the Czech original seems to be, "My parents replied that I AM not at home." Something we just have to learn and get used to, I guess! :-)