Having 60 lines on one plot is likely to be confusing but here is an example. I colored the lines using fatherline as a continuous variable and I made all the lines parallel because that was easy.
library("ggplot2")
library(tidyr)
DF <- data.frame(fatherline = 1:60,
Y1988_Y1989 = seq(0.25, 15, 0.25),
Y1990_Y1991 = seq(1.25, 16, 0.25),
Y1992_Y1993 = seq(2.25, 17, 0.25))
head(DF)
#> fatherline Y1988_Y1989 Y1990_Y1991 Y1992_Y1993
#> 1 1 0.25 1.25 2.25
#> 2 2 0.50 1.50 2.50
#> 3 3 0.75 1.75 2.75
#> 4 4 1.00 2.00 3.00
#> 5 5 1.25 2.25 3.25
#> 6 6 1.50 2.50 3.50
DFtall <- pivot_longer(DF, Y1988_Y1989:Y1992_Y1993, "Year")
head(DFtall)
#> # A tibble: 6 x 3
#> fatherline Year value
#> <int> <chr> <dbl>
#> 1 1 Y1988_Y1989 0.25
#> 2 1 Y1990_Y1991 1.25
#> 3 1 Y1992_Y1993 2.25
#> 4 2 Y1988_Y1989 0.5
#> 5 2 Y1990_Y1991 1.5
#> 6 2 Y1992_Y1993 2.5
ggplot(DFtall, aes(Year, value, group = fatherline, color = fatherline)) + geom_line()

Created on 2020-01-21 by the reprex package (v0.3.0)