For a potential difference V across a resistor R, the current I that flows through the resistor is given by Ohm's law as I = V/R
As current is inversely proportional to resistance, for the same potential difference if the resistance is higher, the current flowing through the resistor is lower. And if the resistance remains the same, for a higher potential difference across the resistor a higher current flows though it.
In a parallel circuit, the potential difference across each of the resistors that make up the circuit is the same. This leads to a higher current flowing through each resistor and subsequently the total current flowing through all the resistors is higher. In a series ciruit on the other hand, the potential difference applied across all the resistors that make up the circuit is divided into smaller fractions across each of the resistors. This leads to a smaller current flowing through each of the resistors and subsequently a smaller total current flowing through the circuit.