Step 1: Understanding the Concept:
Recombinant Inbred Lines (RILs) are homozygous lines developed through repeated selfing or sibling mating of progeny from an initial cross between two parental genotypes.
RIL populations are valuable tools for genetic mapping and quantitative trait loci (QTL) analysis.
Step 2: Detailed Explanation:
Let us analyze the genetics of RIL population development:
The crossing of two homozygous parents (P$_1$ and P$_2$) with different genotypes ($A_1A_1$ and $A_2A_2$) yields heterozygous F$_1$ offspring ($A_1A_2$).
The F$_1$ plants are self-pollinated to produce the F$_2$ generation, which is then subjected to single-seed descent (SSD) and selfing for several generations ($F_6$ to $F_8$).
With each generation of selfing, heterozygosity is reduced by $50\%$, resulting in highly homozygous lines ($A_1A_1$ or $A_2A_2$) at each locus.
Because the RIL population is derived entirely from two parental genotypes, only the two parental alleles ($A_1$ and $A_2$) are present in the gene pool.
No new alleles can be introduced during the selfing process (excluding rare spontaneous mutations).
Therefore, at any given codominant Simple Sequence Repeat (SSR) marker locus, an individual RIL will be homozygous for either the P$_1$ allele or the P$_2$ allele.
The entire RIL population can display a maximum of two parental allele types at that locus.
Step 3: Final Answer:
The maximum number of allele types a biparental RIL population can show at an SSR locus is 2, which corresponds to Option (C).