Research says resistance training can improve distance running performance and/or running economy.
You improve via your body's adaptation to an overload stimilus. If you do not achieve overload and simply repeat this same stimilus habituation eventually occurs. So while simple bodyweight exercise will first provide a substantial stimilus eventually the stimilus will need to be more difficult.
If you're still scared to lift weight then try making your bodyweight exercise more difficult:
Pushup Progression: Pushup--Pushups Feet Elevated--Pushups Feet & Hands Elevated---Plyo Pushups---Explosive Pushups--Pushups Over Medicine Ball
Squat Progression: Squats--Split Squat--Step Up--Bulgarian Split Squat--Squat Jumps--Split Jump--Box Jump
Pullup Progression: Inverted Row--Inverted Row Feet Elevated--Chinup--Pullup--Wide Grip Pullup--Weighted Variations
Dips Progression: Dips Feet on Bench--Regular Dips--Weighted Dips
Alan
Alan