If the parameter to be found is a single number, then you can do brute-force prediction calculations based on a lot of K's (as Geoff suggests), then plot the difference between computational predictions and measured data.
This gives a "complete picture" of how changing the parameter will influence all the measurements, and is not that costly since solving thousands of ODE's takes only a few minutes on modern computers. You can then visually inspect the results to see what K to choose and how sensitive the results are to changes in K - ie., how certain/uncertain you should be about your result.
At this point, you can do some simple least-squares or statistical analysis on the results if you want, but you'll probably get a better idea of what's going on by just looking at it.
Here's some code to do the brute force calculations and plotting! I release this code to the public domain.
% Some randomish input data (replace with your times tt and function f)
Ktrue = 2.31;
ff = ifft(fft(exp(randn(1,length(tt)))).* ...
ff = ff/mean(ff);
f = @(t) interp1(tt,ff,t);
% manufacturing state data X1,X2 (replace with your X1,X2)
X1initial = 0.93;
X2initial = 1.87;
odefuntrue = @(t,Y) [f(t)*Y(1)*Y(2), -Ktrue*f(t)*Y(1)*Y(2)]';
[T,Ytrue] = ode45(odefuntrue,tt,[X1initial,X2initial]);
X1 = Ytrue(:,1);
X2 = Ytrue(:,2);
% Solve the ODE for a lot of different values of K
Ts = zeros(length(tt),length(K));
X1s = Ts; X2s = Ts;
odefun = @(t,Y) [f(t)*Y(1)*Y(2), -K(jj)*f(t)*Y(1)*Y(2)]';
[T,Y] = ode45(odefun,tt,[X1(1),X2(1)]);
% Compute the difference between the measured state, (X1,X2), and
% the predicted state for all different K's, (X1s,X2s)
errors1 = X1*ones(1,length(K)) - X1s;
errors2 = X2*ones(1,length(K)) - X2s;
% Plot the full errors at all times for different values of K
title('error in X1');
title('error in X2');
And here's the results. You can see how the error is zero along the vertical line $K=2.31$ corresponding to the true $K$, and the horizontal line $t=0$, and becomes nonzero as $K$ changes away from the true value and time increases.