Can I run the wires from a cable into a panel through separate knockouts?
No never; you can never split a cable like that.  

What you can do is transition to EMT conduit and individual THHN wires, which will pack nicely in EMT conduit.

I would start by heading to a proper electrical supply house for a cable clamp that is a proper fit for your large cable.  You need a proper electrical supply because they have both the depth of stock and knowledge.  Home Depot etc. will sell you the wrong thing with a smile. 

Next, I would buy a large steel box, say 4-11/16" square deep, or 6" square, with knockouts that will fit that cable clamp.  

I would use EMT metal conduit to plumb out of a knockout on the service panel, to a convenient location. This can be just a foot or two.  There, install the box. Use the largest size EMT the panel knockout can accept (3/4", surely)  and correct fittings. If the junction box hole is too large, they make special washers to solve that, but then,  you can't use the EMT as ground path, so run a #10 ground wire. 

There should be a #10-32 tapped hole in the back of the box for a grounding screw. If you are wiring a ground wire through the EMT, pigtail off this. 

Then, you terminate your cable in the large junction box in the normal way (give yourself 12" of wire length).  Terminate the ground wire to the box chassis or other grounds. 

For each of the hot and neutral wires, obtain the appropriate gauge copper THHN/THWN-2 wires with enough length to reach the breaker, run through the EMT and give 12" spare length in the box.  You need one white or gray for the neutral, and two of any other color but green.  Black and black is fine. 

Run the three individual THHN wires through the EMT conduit, terminate one end at the breaker or neutral bar.  Inside the box tie the individual wires to the wires of the cable. 

Three THHN copper wires of #6 will certainly fit in a 3/4" EMT conduit, and are good for 70A. This is more than #6 cable.  

If the wires are small enough, you can use large wire nuts for the splice, otherwise use right-sized Polaris style connectors. If your cable is aluminum, that is fine but you must use Polaris.