Here is one fairly simple way to find the parameters. The two equations are
dhare/dt = k1*hare - k2*hare*lynx
dlynx/dt = -k3*lynx + k2*hare*lynx
The steady state fixed point occurs when both time derivatives are zero in which case the fixed point is at [hare0,lynx0] and you can solve and get the standard result
hare0 = k3/k2 lynx0 = k1/k2
For Lotka-Volterra, the oscillations form a closed single-loop orbit. Fortunately for the Hudson Bay data someone chose to provide 21 years so there is an almost exactly closed two-loop curve. (The loops would be identical for perfect L-V behavior but of course this is real data).
Suppose 'hare' is a row vector of population by year. I added the first point to the end of the array with hare = [hare hare(1)] (similarly for lynx) so now if you plot hare vs lynx you will get a closed curve with two loops.
So far so good. Now suppose you time integrate both sides of the first L-V equation around an integral number of orbits, going from tstart to tfin. On the left hand side, integrating the derivative you get hare(tstart)-hare(tfin) = 0 since the start position and the end position are the same. On the right hand side you get
0 = k1*Ihare - k2*Iboth where Ihare = Int hare dt and Iboth = Int hare*lynx dt
Iboth/Ihare = k1/k2 = lynx0
and similarly for the second equation
Iboth/Ilynx = k3/k2 = hare0
For the integrations you can use t = 0:21 since you have one-year intervals and use the trapz function ,
Ihare = trapz(t,hare) (similary for lynx); Iboth = trapz(t,hare.*lynx)
I got lynx0 = 2.0883e+04 hare0 = 3.5292e+04
which, given the look of the data, ought to be rounded to something like 21000 and 35000. Once you have those, with a new k1 you can use the eqns above to solve for k2 and k3.
A long time ago there was a comic strip about Fat Freddy's Cat. One day Fat Freddy's Cat went out toward the woods and met a big kind of cat he had never seen before. He said, what kind of cat are you? And the cat said, "I'm a lynx". To which FFC said, "Oh, I always wondered where links sausage came from".