Why do Allylic Halides prefer SN2 reaction over SN1?
You and your instructor are both correct, it is your premise that's wrong.

Your instructor's explanation about the allyl double bond stabilizing the $\mathrm{S_{N}2}$ transition state is correct. The following figure shows the stabilizing interaction.



image source

If we remove the attacking nucleophile and the leaving group and place a positive charge in the system, then we have the $\mathrm{S_{N}1}$ allyl intermediate.  As you point out, this intermediate is resonance stabilized, and therefore, the transition state leading to it is also stabilized.

Your premise,


  Why do Allylic Halides prefer SN2 reaction over Sn1?


is not true.  There are many cases where allylic halides react preferentially by an $\mathrm{S_{N}1}$ process.

Since both the allylic $\mathrm{S_{N}1}$ and $\mathrm{S_{N}2}$ reactions are stabilized, there is a delicate balance between the two pathways.  We can shift from one mechanism to the other by changing reaction conditions.  

For example, allyl chloride reacts by an $\mathrm{S_{N}1}$ mechanism to produce allyl alcohol when we place it in a 50:50 mix of $\ce{H2O:EtOH}$ at 45°C.  If instead we run the reaction in the presence of ethoxide ion in ethanol at 45°C, the corresponding ethyl ether is formed by an $\mathrm{S_{N}2}$ process.