The diophantine equation that needs to be solved is $kx^2-y^2=3$ with the added conditions that:
\begin{cases}k>1\\k\text{ not a square}\\ x>2\\ x\equiv 0\mod 2\\ x/2 \not| \ k \\ x/2 \text{ is prime}\\y\ge 2a_0-1\ \text{for a given $a_0>1$ (If the general case is difficult, you can take $y\ge 3$)}\\ y\equiv 1\mod 2 \end{cases}
Does there exist solutions to this?
Is there a possibility of finding a parametrization of the solution space, given that there exists solutions?
On
You also have some computations by Dmitry; I suspect he put all your conditions into his program. I am just showing parametrized families with fixed $k$ of $w^2 - k v^2 = -3.$ The parametrization comes from first solving $r^2 - k s^2 = 1,$ using the smallest solution with $r,s > 0.$ When $k$ is not divisible by $3,$ there are two interlaced families, that is the quick description of the output saying "BACK ONE STEP" Ummm; given $w^2 - k v^2 = -3,$ the "next" solution, either one or two lines after, comes from $$ (w,v) \mapsto (rw+ksv, \; sw + rv) $$
anyway these are the values of $k$ below 29 that work..
w^2 - 3 v^2 = -3 = -3
Sat Apr 4 11:17:33 PDT 2020
w: 3 v: 2 SEED BACK ONE STEP 0 , 1
w: 12 v: 7
w: 45 v: 26
w: 168 v: 97
w: 627 v: 362
w: 2340 v: 1351
w: 8733 v: 5042
w: 32592 v: 18817
w: 121635 v: 70226
w: 453948 v: 262087
w: 1694157 v: 978122
w: 6322680 v: 3650401
Sat Apr 4 11:18:04 PDT 2020
w^2 - 3 v^2 = -3 = -3
====================================================
w^2 - 7 v^2 = -3 = -3
Sat Apr 4 11:20:31 PDT 2020
w: 2 v: 1 SEED KEEP +-
w: 5 v: 2 SEED BACK ONE STEP -2 , 1
w: 37 v: 14
w: 82 v: 31
w: 590 v: 223
w: 1307 v: 494
w: 9403 v: 3554
w: 20830 v: 7873
w: 149858 v: 56641
w: 331973 v: 125474
w: 2388325 v: 902702
w: 5290738 v: 1999711
Sat Apr 4 11:20:52 PDT 2020
w^2 - 7 v^2 = -3 = -3
====================================================
w^2 - 12 v^2 = -3 = -3
Sat Apr 4 11:26:19 PDT 2020
w: 3 v: 1 SEED KEEP +-
w: 45 v: 13
w: 627 v: 181
w: 8733 v: 2521
w: 121635 v: 35113
w: 1694157 v: 489061
Sat Apr 4 11:26:34 PDT 2020
w^2 - 12 v^2 = -3 = -3
=============================================
w^2 - 13 v^2 = -3 = -3
Sat Apr 4 11:28:01 PDT 2020
w: 7 v: 2 SEED KEEP +-
w: 137 v: 38 SEED BACK ONE STEP -7 , 2
w: 9223 v: 2558
w: 177833 v: 49322
w: 11971447 v: 3320282
Sat Apr 4 11:28:16 PDT 2020
w^2 - 13 v^2 = -3 = -3
=============================================
w^2 - 19 v^2 = -3 = -3
Sat Apr 4 11:30:43 PDT 2020
w: 4 v: 1 SEED KEEP +-
w: 61 v: 14 SEED BACK ONE STEP -4 , 1
w: 1421 v: 326
w: 20744 v: 4759
w: 483136 v: 110839
w: 7052899 v: 1618046
Sat Apr 4 11:30:58 PDT 2020
w^2 - 19 v^2 = -3 = -3
===============================================
w^2 - 21 v^2 = -3 = -3
Sat Apr 4 11:32:41 PDT 2020
w: 9 v: 2 SEED KEEP +-
w: 999 v: 218
w: 109881 v: 23978
w: 12085911 v: 2637362
Sat Apr 4 11:32:56 PDT 2020
w^2 - 21 v^2 = -3 = -3
==================================================
w^2 - 28 v^2 = -3 = -3
Sat Apr 4 11:35:32 PDT 2020
w: 5 v: 1 SEED KEEP +-
w: 37 v: 7 SEED BACK ONE STEP -5 , 1
w: 1307 v: 247
w: 9403 v: 1777
w: 331973 v: 62737
w: 2388325 v: 451351
Sat Apr 4 11:35:47 PDT 2020
w^2 - 28 v^2 = -3 = -3
=========================================================
Some first solutions
(k,x,y):gp-code: