Tri-Band Spark phones do not currently have the ability to use data while on a call because there is only one radio which handles everything. So, if you are on CDMA/3G, you cannot be on LTE. It won't be until Sprint offers VoLTE or similar will you be able to do this again.
The way Sprint now handles routing is by using a technology called CSFB/eCSFB (Circuit-Switched Fall-Back). When you receive a call, your phone is told to switch back to CDMA to accept the call. In an upgraded area that is correctly configured and operating, this should be nearly instant bar any technical issues. This is all handled on the network side and there's nothing a consumer can do to circumvent it outside of forcing the phone to operate in LTE mode at all times. Though, this will not allow you to receive phone calls.
HD Voice on the other hand, relies on the devices on both ends of the call as well as the network. Both phones must be HD Voice compatible for this to operate. This supposedly offers better noise cancellation and clarity.