Why does the capacitor start to discharge from peak and not from V(input<0) in a peak detector?
Once C1 becomes charged to the peak voltage, it can only receive charge again when: -


the input voltage rises to a higher peak voltage
the capacitor is discharged by the parallel resistor and another  input peak comes along


So, if your input voltage peaked once and then went to zero volts forever, the capacitor (C1) will receive that peak input voltage and then will be slowly discharged by R1.