Phono Pre Upgrade from Parasound JC-3?

I tend to agree with @mijostyn that you will have a hard time improving on the JC-3 at this price level, especially if balanced operation is required.

The Boulder 1008 would be a major step up, which I've compared head-to-head with the JC-3. Retail it's outside of your budget, but you might try and find one on the used market for around 6k. 

You can also consider Boulder's new 508, which apparently comes pretty close to the 1008 and retails for around 6k. However, it doesn't offer any flexibility with a fixed output at 66dB and a fixed 100 ohm load impedance. While these settings work well for most low out, low impedance MC cartridges, it may not suit your requirements.

@lewm I use the balanced cables between the JC3+ and a Lyngdorf TDAI 3400 amp, which is also balanced.   

If you are MC, Rega Aura. I’ve had it for 2 days and it blows my mind and can’t wait to get it broken in. $6k.  Has RCA and balanced.  Mute button, mono switch,  and gain of 63.5 db and 69.5 db.  You can switch capacitance and loading on the fly, just mute it first. 

Dear @snackeyp  : """  as I like the warmer sound of tubes... """

so then that's what you like not MUSIC because MUSIC is all you want it but warm.

Good luck with your warm BAT. Btw, the worst place to have tubes is at the phono audio system link.

Regards and enjoy the MUSIC NOT DISTORTIONS,
R.

Pass Labs XP-17 or if you can the Pass XP-27