Inductors in Series and Parallel
In series, flux linkages add and inductance grows
To L1 = 4mH a second inductor L2 is connected in series, and moving L2 changes its value. The top bar is the series inductance L_s = L1 + L2, with the darker segment for the first inductor and the lighter one for the second. The green bar below is the inductance of the same two inductors in parallel, and however large the second one gets, it never grows past the darker segment of the top bar. Set the series inductance to the target of 10mH on the orange dashed line by adjusting L2.
In series, flux linkages add
Connect inductors in series and the same current I passes through both coils in turn. A coil's flux linkage λ is proportional to its current (λ = LI), so the linkages of the two coils simply add: λ = L1·I + L2·I = (L1 + L2)·I. When the current changes, the back emf the two coils produce together, V = L1·dI/dt + L2·dI/dt, adds in the same way. So the series inductance is L_s = L1 + L2, the sum of the two, and that addition is also why the top bar looks like the darker and lighter segments joined end to end.
In parallel, reciprocals add
Split inductors in parallel and the same voltage V sits across both coils while the current divides between two branches. From V = L·dI/dt, the rates of change of the branch currents are dI1/dt = V/L1 and dI2/dt = V/L2, and since the total current is their sum, dI/dt = V/L1 + V/L2 = V·(1/L1 + 1/L2). Matching this with V = L_p·dI/dt gives 1/L_p = 1/L1 + 1/L2, that is, L_p = L1·L2/(L1+L2), so the parallel inductance is always smaller than even the smaller of the two. At the same voltage both branch currents rise together, so the total current changes faster, and from outside the circuit it looks like a coil with that much less inductance.
On the same side as resistors
Resistors add in series and their reciprocals add in parallel, and inductors do exactly the same. Only capacitors are reversed, adding in parallel and adding reciprocals in series. When it blurs, picture the shared current: in series the same current passes through both coils, so the opposing voltages stack up and the inductance grows. This addition rule holds when the flux of one coil does not link the other; place the coils close enough to create mutual inductance and that share is added or subtracted.
Back to the first screen
As you raised L2, the series bar on top grew by exactly that much, and it reached the target of 10mH when L2 was 6mH (L1 = 4). Even then the parallel bar below was shorter than the darker segment of the top bar, showing plainly that splitting in parallel gives less inductance than even the smaller coil. Series grows because the same current runs through both and the flux linkages add, and parallel shrinks because at the same voltage both branch currents rise together, so the reciprocals add. It is the mirror opposite of capacitors, yet exactly like resistors.