We could use R:

# example data
df1 <- read.table(text = "
Individual  SNP1    SNP2    SNP3    SNP4
Father_1    1   0   1   2
Mother_1    0   0   2   1
Father_2    1   0   1   1
Mother_2    2   1   0   1
Father_3    1   2   0   1
Mother_3    1   .   0   2
", header = TRUE, na.string = ".")

Using loops

t(sapply(split(df1[, -1], rep(1:(nrow(df1)/2), each = 2)), colMeans))
#   SNP1 SNP2 SNP3 SNP4
# 1  0.5  0.0  1.5  1.5
# 2  1.5  0.5  0.5  1.0
# 3  1.0   NA  0.0  1.5

Or

# subset using recycling then get mean
(df1[ c(TRUE, FALSE), -1] + df1[ c(FALSE, TRUE), -1]) / 2

Using awk:

# get odd rows
sed -n 'p;n' original.txt > file1.txt
# get even rows
sed -n 'n;p' original.txt > file2.txt

Then use the awk script from SO: Average of multiple files in shell:

awk 'FNR == 1 { nfiles++; ncols = NF }
     { for (i = 1; i < NF; i++) sum[FNR,i] += $i
       if (FNR > maxnr) maxnr = FNR
     }
     END {
         for (line = 1; line <= maxnr; line++)
         {
             for (col = 1; col < ncols; col++)
                  printf "  %f", sum[line,col]/nfiles;
             printf "\n"
         }
     }' file*.txt

In python you can do this with reading the file two lines at a time, so you never hold more than two lines at once. This will use next to no memory, and I'm pretty sure it will be much faster than the R solution. There is probably also a solution with pandas and chunking the input file read.

infile = open("infile.tsv")
header = infile.readline()
header = header.strip().split()[1:]
print("\t".join(header))

while True:

    line1 = infile.readline()
    line2 = infile.readline()

    if (not line1) or (not line2): break # EOF

    assert "Father" in line1 and "Mother" in line2

    genotypes1 = map(float, line1.strip().split()[1:])
    genotypes2 = map(float, line2.strip().split()[1:])

    average_genotypes = [str((x+y)/2) for x, y in zip(genotypes1, genotypes2)]

    print("\t".join(average_genotypes))